# NAG Library Contents Overview

## Chapter Summaries

### A00 – Library Identification

 Full details of this chapter for your environment: Fortran C Toolbox .NET

The routines in this chapter provide information about the NAG Library.

Information about the precise implementation of the NAG Library in use will be needed when communicating with the NAG Technical Support Service (see NAG Library Manual Introductory document 'Support from NAG').

### A02 – Complex Arithmetic

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter provides facilities for arithmetic operations involving complex numbers.

### C02 – Zeros of Polynomials

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with computing the zeros of a polynomial with real or complex coefficients.

### C05 – Roots of One or More Transcendental Equations

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the calculation of zeros of continuous functions of one or more variables. The majority of problems considered are for real-valued functions of real variables, in which case complex equations must be expressed in terms of the equivalent larger system of real equations.

### C06 – Summation of Series

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the following tasks.

• Calculating the discrete Fourier transform of a sequence of real or complex data values.
• Calculating the discrete convolution or the discrete correlation of two sequences of real or complex data values using discrete Fourier transforms.
• Calculating the inverse Laplace transform of a user-supplied routine.
• Calculating the fast Gauss transform approximation to the discrete Gauss transform.
• Direct summation of orthogonal series.
• Acceleration of convergence of a sequence of real values.

### C09 – Wavelet Transforms

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the analysis of datasets (or functions or operators) in terms of frequency and scale components using wavelet transforms. Wavelet transforms have been applied in many fields from time series analysis to image processing and the localization in either frequency or scale that they provide is useful for data compression or denoising. In general the standard wavelet transform uses dilation and scaling of a chosen function, ψ(t), (called the mother wavelet) such that ψa,b(t)=1/(sqrt(a))ψ((t-b)/a) where a gives the scaling and b determines the translation. Wavelet methods can be divided into continuous transforms and discrete transforms. In the continuous case, the pair a and b are real numbers with a>0. For the discrete transform, a and b can be chosen as a=2-j, b=k2-j for integers j, k ψj,k(t)=2j/2ψ(2jt-k).

The continuous real valued, one-dimensional wavelet transform (CWT) is included in this chapter. The discrete wavelet transform (DWT) at a single level together with its inverse and the multi-level DWT with inverse are also provided for one, two and three dimensions. The Maximal Overlap DWT (MODWT) together with its inverse and the multi-level MODWT with inverse are provided for one dimension. The choice of wavelet for CWT includes the Morlet wavelet and derivatives of a Gaussian while the DWT and MODWT offer the orthogonal wavelets of Daubechies and a selection of biorthogonal wavelets.

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides routines for the numerical evaluation of definite integrals in one or more dimensions and for evaluating weights and abscissae of integration rules.

### D02 – Ordinary Differential Equations

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with the numerical solution of ordinary differential equations. There are two main types of problem: those in which all boundary conditions are specified at one point (initial value problems), and those in which the boundary conditions are distributed between two or more points (boundary value problems and eigenvalue problems). Routines are available for initial value problems, two-point boundary value problems and Sturm–Liouville eigenvalue problems.

### D03 – Partial Differential Equations

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with the numerical solution of partial differential equations.

### D04 – Numerical Differentiation

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with calculating approximations to derivatives of a function f.

### D05 – Integral Equations

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with the numerical solution of integral equations. Provision will be made for most of the standard types of equation (see the Chapter Introduction). The following are, however, specifically excluded:

• Equations arising in the solution of partial differential equations by integral equation methods. In cases where the prime purpose of an algorithm is the solution of a partial differential equation it will normally be included in Chapter D03.
• Calculation of inverse integral transforms. This problem falls within the scope of Chapter C06.

### D06 – Mesh Generation

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with automatic mesh generation

• with line segments, over the boundary of a closed two-dimensional connected polygonal domain;
• with triangles, over a given two-dimensional region using only its boundary mesh.

### E01 – Interpolation

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the interpolation of a function of one or more variables. When provided with the value of the function (and possibly one or more of its lowest-order derivatives) at each of a number of values of the variable(s), the NAG Library routines provide either an interpolating function or an interpolated value. For some of the interpolating functions, there are supporting NAG Library routines to evaluate, differentiate or integrate them.

### E02 – Curve and Surface Fitting

 Full details of this chapter for your environment: Fortran C Toolbox .NET

The main aim of this chapter is to assist you in finding a function which approximates a set of data points. Typically the data contain random errors, as of experimental measurement, which need to be smoothed out. To seek an approximation to the data, it is first necessary to specify for the approximating function a mathematical form (a polynomial, for example) which contains a number of unspecified coefficients: the appropriate fitting routine then derives for the coefficients the values which provide the best fit of that particular form. The chapter deals mainly with curve and surface fitting (i.e., fitting with functions of one and of two variables) when a polynomial or a cubic spline is used as the fitting function, since these cover the most common needs. However, fitting with other functions and/or more variables can be undertaken by means of general linear or nonlinear routines (some of which are contained in other chapters) depending on whether the coefficients in the function occur linearly or nonlinearly. Cases where a graph rather than a set of data points is given can be treated simply by first reading a suitable set of points from the graph.

The chapter also contains routines for evaluating, differentiating and integrating polynomial and spline curves and surfaces, once the numerical values of their coefficients have been determined.

There is also a routine for computing a Padé approximant of a mathematical function (see the Chapter Introduction).

### E04 – Minimizing or Maximizing a Function

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides routines for solving various mathematical optimization problems by solvers based on local stopping criteria. The main classes of problems covered in this chapter are:

• Linear Programming (LP) – dense and sparse;
• Quadratic Programming (QP) – convex and nonconvex, dense and sparse;
• Nonlinear Programming (NLP) – dense and sparse, based on active-set SQP methods or interior point methods (IPM);
• Semidefinite Programming (SDP) – both linear matrix inequalities (LMI) and bilinear matrix inequalities (BMI);
• Derivative-free Optimization (DFO);
• Least Squares (LSQ), data fitting – linear and nonlinear, constrained and unconstrained.

For a full overview of the functionality offered in this chapter, see the Chapter Introduction or the Chapter Contents (Chapter E04).

• Chapter E05 contains routines to solve global optimization problems;
• Chapter H addresses problems arising in operational research and focuses on Mixed Integer Programming (MIP);
• Chapters F07 and F08 include routines for linear algebra and in particular unconstrained linear least squares;
• Chapter E02 focuses on curve and surface fitting, in which linear data fitting in l1 or l norm might be of interest.

This introduction is only a brief guide to the subject of optimization. It discusses a classification of the optimization problems and presents an overview of the algorithms and their stopping criteria to help with the choice of a correct solver for a particular problem. Anyone with a difficult or protracted problem to solve will find it beneficial to consult a more detailed text, see the References section in the Chapter Introduction. If you are unfamiliar with the mathematics of the subject you may find the Chapter Introduction a useful starting point.

### E05 – Global Optimization of a Function

 Full details of this chapter for your environment: Fortran C Toolbox .NET

Global optimization involves finding the absolute maximum or minimum value of a function (the objective function) of several variables, possibly subject to restrictions (defined by a set of bounds or constraint functions) on the values of the variables. Such problems can be much harder to solve than local optimization problems (which are discussed in Chapter E04) because it is difficult to determine whether a potential optimum found is global, and because of the nonlocal methods required to avoid becoming trapped near local optima. Most optimization routines in the NAG Library are concerned with function minimization only, since the problem of maximizing a given objective function F is equivalent to minimizing -F. In e05jb, e05sa and e05sb, you may specify whether you are solving a minimization or maximization problem; in the latter case, the required transformation of the objective function will be carried out automatically. In what follows we refer exclusively to minimization problems.

This introduction is a brief guide to the subject of global optimization, designed for the casual user. For further details you may find it beneficial to consult a more detailed text, see the References section in the Chapter Introduction. Furthermore, much of the material in the E04 Chapter Introduction is also relevant in this context and it is strongly recommended that you read the E04 Chapter Introduction.

### F01 – Matrix Operations, Including Inversion

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides facilities for four types of problem:

• Matrix Inversion
• Matrix Factorizations
• Matrix Arithmetic and Manipulation
• Matrix Functions

See the Chapter Introduction where these problems are discussed.

### F02 – Eigenvalues and Eigenvectors

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter provides routines for various types of matrix eigenvalue problem:

• standard eigenvalue problems (finding eigenvalues and eigenvectors of a square matrix A);
• singular value problems (finding singular values and singular vectors of a rectangular matrix A);
• generalized eigenvalue problems (finding eigenvalues and eigenvectors of a matrix pencil A-λB).
• quadratic eigenvalue problems (finding eigenvalues and eigenvectors of the quadratic λ2A+λB+C).

Routines are provided for both real and complex data.

The majority of routines for these problems can be found in Chapter F08 which contains software derived from LAPACK (see Anderson et al. (1999) LAPACK Users' Guide). However, you should read the F02 Chapter Introduction before turning to Chapter F08, especially if you are a new user. Chapter F12 contains routines for large sparse eigenvalue problems, although one such routine is also available in this chapter.

Chapters F02 and F08 contain Black Box (or Driver) routines that enable many problems to be solved by a call to a single routine, and the decision trees in the Chapter Introduction direct you to the most appropriate routines in Chapters F02 and F08. The Chapter F02 routines call routines in Chapters F07 and F08 wherever possible to perform the computations, and there are pointers in the Chapter Introduction to the relevant decision trees in Chapter F08.

### F03 – Determinants

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with the calculation of determinants of square matrices.

### F04 – Simultaneous Linear Equations

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with the solution of the matrix equation AX=B, where B may be a single vector or a matrix of multiple right-hand sides. The matrix A may be real, complex, symmetric, Hermitian, positive definite, positive definite Toeplitz or banded. It may also be rectangular, in which case a least squares solution is obtained.

Much of the functionality of this chapter has been superseded by routines from Chapters F07 and F08 (LAPACK routines) as those chapters have grown and have included driver and expert driver routines.

For a general introduction to sparse systems of equations, see the F11 Chapter Introduction, which provides routines for large sparse systems. Some routines for sparse problems are also included in this chapter; they are described in the Chapter Introduction.

### F05 – Orthogonalization

 Full details of this chapter for your environment: Fortran Toolbox

This chapter is concerned with the orthogonalization of vectors in a finite dimensional space.

### F06 – Linear Algebra Support Routines

 Full details of this chapter for your environment: Fortran C .NET

This chapter is concerned with basic linear algebra routines which perform elementary algebraic operations involving scalars, vectors and matrices. It includes routines which conform to the specifications of the BLAS (Basic Linear Algebra Subprograms).

### F07 – Linear Equations (LAPACK)

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides routines for the solution of systems of simultaneous linear equations, and associated computations. It provides routines for

• matrix factorizations;
• solution of linear equations;
• estimating matrix condition numbers;
• computing error bounds for the solution of linear equations;
• matrix inversion;
• computing scaling factors to equilibrate a matrix.

Routines are provided for both real and complex data.

For a general introduction to the solution of systems of linear equations, you should turn first to the F04 Chapter Introduction. The decision trees, in the F04 Chapter Introduction, direct you to the most appropriate routines in Chapters F04 and F07 for solving your particular problem. In particular, Chapters F04 and F07 contain Black Box (or driver) routines which enable some standard types of problem to be solved by a call to a single routine. Where possible, routines in Chapter F04 call Chapter F07 routines to perform the necessary computational tasks.

There are two types of driver routines in this chapter: simple drivers which just return the solution to the linear equations; and expert drivers which also return condition and error estimates and, in many cases, also allow equilibration. The simple drivers for real matrices have names of the form F07_AF (D__SV) and for complex matrices have names of the form F07_NF (Z__SV). The expert drivers for real matrices have names of the form F07_BF (D__SVX) and for complex matrices have names of the form F07_PF (Z__SVX).

The routines in this chapter (Chapter F07) handle only dense and band matrices (not matrices with more specialised structures, or general sparse matrices).

The routines in this chapter have all been derived from the LAPACK project (see Anderson et al. (1999) LAPACK Users' Guide). They have been designed to be efficient on a wide range of high-performance computers, without compromising efficiency on conventional serial machines.

### F08 – Least Squares and Eigenvalue Problems (LAPACK)

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides routines for the solution of linear least squares problems, eigenvalue problems and singular value problems, as well as associated computations. It provides routines for:

• solution of linear least squares problems
• solution of symmetric eigenvalue problems
• solution of nonsymmetric eigenvalue problems
• solution of singular value problems
• solution of generalized linear least squares problems
• solution of generalized symmetric-definite eigenvalue problems
• solution of generalized nonsymmetric eigenvalue problems
• solution of generalized singular value problems
• matrix factorizations associated with the above problems
• estimating condition numbers of eigenvalue and eigenvector problems
• estimating the numerical rank of a matrix
• solution of the Sylvester matrix equation

Routines are provided for both real and complex data.

For a general introduction to the solution of linear least squares problems, you should turn first to Chapter F04. The decision trees, at the end of Chapter F04, direct you to the most appropriate routines in Chapters F04 and F08. Chapters F04 and F08 contain Black Box (or driver) routines which enable standard linear least squares problems to be solved by a call to a single routine.

For a general introduction to eigenvalue and singular value problems, you should turn first to Chapter F02. The decision trees, at the end of Chapter F02, direct you to the most appropriate routines in Chapters F02 and F08. Chapters F02 and F08 contain Black Box (or driver) routines which enable standard types of problem to be solved by a call to a single routine. Often routines in Chapter F02 call Chapter F08 routines to perform the necessary computational tasks.

The routines in this chapter (Chapter F08) handle only dense, band, tridiagonal and Hessenberg matrices (not matrices with more specialised structures, or general sparse matrices). The tables in the Chapter Introduction and the decision trees in the Chapter Introduction direct you to the most appropriate routines in Chapter F08.

The routines in this chapter have all been derived from the LAPACK project (see Anderson et al. (1999) LAPACK Users' Guide). They have been designed to be efficient on a wide range of high-performance computers, without compromising efficiency on conventional serial machines.

### F11 – Large Scale Linear Systems

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter provides routines for the solution of large sparse systems of simultaneous linear equations. These include iterative methods for real nonsymmetric and symmetric, complex non-Hermitian and Hermitian linear systems and direct methods for general real linear systems. Further direct methods are currently available in Chapters F01 and F04.

### F12 – Large Scale Eigenproblems

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter provides routines for computing some eigenvalues and eigenvectors of large-scale (sparse) standard and generalized eigenvalue problems. It provides routines for:

• solution of symmetric eigenvalue problems;
• solution of nonsymmetric eigenvalue problems;
• solution of generalized symmetric-definite eigenvalue problems;
• solution of generalized nonsymmetric eigenvalue problems;
• partial singular value decomposition.

Routines are provided for both real and complex data.

The routines in this chapter have all been derived from the ARPACK software suite (see Lehoucq et al. (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods), a collection of Fortran 77 routines designed to solve large scale eigenvalue problems. The interfaces provided in this chapter have been chosen to combine ease of use with the flexibility of the original ARPACK software. The underlying iterative methods and algorithms remain essentially the same as those in ARPACK and are described fully in Lehoucq et al. (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods.

The algorithms used are based upon an algorithmic variant of the Arnoldi process called the Implicitly Restarted Arnoldi Method. For symmetric matrices, this reduces to a variant of the Lanczos process called the Implicitly Restarted Lanczos Method. These variants may be viewed as a synthesis of the Arnoldi/Lanczos process with the Implicitly Shifted QR technique that is suitable for large scale problems. For many standard problems, a matrix factorization is not required. Only the action of the matrix on a vector is needed.

### F16 – Further Linear Algebra Support Routines

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with basic linear algebra routines which perform elementary algebraic operations involving scalars, vectors and matrices. Most routines for such operations conform either to the specifications of the BLAS (Basic Linear Algebra Subprograms) or to the specifications of the BLAST (Basic Linear Algebra Subprograms Technical) Forum. This chapter includes routines from the BLAST specifications. Most (BLAS) routines for such operations are available in Chapter F06.

### G01 – Simple Calculations on Statistical Data

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter covers three topics:

• plots, descriptive statistics, and exploratory data analysis;
• statistical distribution functions and their inverses;
• testing for Normality and other distributions.

### G02 – Correlation and Regression Analysis

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with two techniques

• correlation analysis and
• regression modelling,
both of which are concerned with determining the inter-relationships among two or more variables.

Other chapters of the NAG Library which cover similar problems are Chapters E02 and E04. Chapter E02 routines may be used to fit linear models by criteria other than least squares, and also for polynomial regression; Chapter E04 routines may be used to fit nonlinear models and linearly constrained linear models.

### G03 – Multivariate Methods

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with methods for studying multivariate data. A multivariate dataset consists of several variables recorded on a number of objects or individuals. Multivariate methods can be classified as those that seek to examine the relationships between the variables (e.g., principal components), known as variable-directed methods, and those that seek to examine the relationships between the objects (e.g., cluster analysis), known as individual-directed methods.

Multiple regression is not included in this chapter as it involves the relationship of a single variable, known as the response variable, to the other variables in the dataset, the explanatory variables. Routines for multiple regression are provided in Chapter G02.

### G04 – Analysis of Variance

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with methods for analysing the results of designed experiments. The range of experiments covered include:

• single factor designs with equal sized blocks such as randomized complete block and balanced incomplete block designs,
• row and column designs such as Latin squares, and
• complete factorial designs.

Further designs may be analysed by combining the analyses provided by multiple calls to routines or by using general linear model routines provided in Chapter G02.

### G05 – Random Number Generators

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the generation of sequences of independent pseudorandom and quasi-random numbers from various distributions, and models.

### G07 – Univariate Estimation

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter deals with the estimation of unknown parameters of a univariate distribution. It includes both point and interval estimation using maximum likelihood and robust methods.

### G08 – Nonparametric Statistics

 Full details of this chapter for your environment: Fortran C Toolbox

The routines in this chapter perform nonparametric statistical tests which are based on distribution-free methods of analysis. For convenience, the chapter contents are divided into five types of test: tests of location, tests of dispersion, tests of distribution, tests of association and correlation, and tests of randomness. There are also routines to fit linear regression models using the ranks of the observations.

The emphasis in this chapter is on testing; if you wish to compute nonparametric correlations you are referred to Chapter G02, which contains several routines for that purpose.

There are a large number of nonparametric tests available. A selection of some of the more commonly used tests are included in this chapter.

### G10 – Smoothing in Statistics

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with methods for smoothing data. Included are methods for density estimation, smoothing time series data, and statistical applications of splines. These methods may also be viewed as nonparametric modelling.

### G11 – Contingency Table Analysis

 Full details of this chapter for your environment: Fortran C Toolbox

The routines in this chapter are for the analysis of discrete multivariate data. One suite of routines computes tables while other routines are for the analysis of two-way contingency tables, conditional logistic models and one-factor analysis of binary data.

Routines in Chapter G02 may be used to fit generalized linear models to discrete data including binary data and contingency tables.

### G12 – Survival Analysis

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with statistical techniques used in the analysis of survival/reliability/failure time data.

Other chapters contain routines which are also used to analyse this type of data. Chapter G02 contains generalized linear models, Chapter G07 contains routines to fit distribution models, and Chapter G08 contains rank based methods.

### G13 – Time Series Analysis

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides facilities for investigating and modelling the statistical structure of series of observations collected at points in time. The models may then be used to forecast the series.

The chapter covers the following models and approaches.

• Univariate time series analysis, including autocorrelation functions and autoregressive moving average (ARMA) models.
• Univariate spectral analysis.
• Transfer function (multi-input) modelling, in which one time series is dependent on other time series.
• Bivariate spectral methods including coherency, gain and input response functions.
• Vector ARMA models for multivariate time series.
• Kalman filter models (linear and nonlinear).
• GARCH models for volatility.
• Inhomogeneous Time Series.

### G22 – Linear Model Specification

 Full details of this chapter for your environment: Fortran C

The routines in this chapter provide a mechanism for specifying a linear model using a text based modelling language and are intended to be used in conjunction with the model fitting routines from other chapters, for example Chapter G02.

### H – Operations Research

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter provides routines to solve certain integer programming, transportation and shortest path problems. Additionally 'best subset' routines are included.

### M01 – Sorting and Searching

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter is concerned with sorting and searching numeric or character data. It handles only the simplest types of data structure and it is concerned only with internal sorting and searching – that is, sorting and searching a set of data which can all be stored within the program.

If you have large files of data or complicated data structures to be sorted or searched you should use a comprehensive sorting or searching program or package.

### S – Approximations of Special Functions

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the provision of some commonly occurring physical and mathematical functions.

### X01 – Mathematical Constants

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with the provision of mathematical constants required by other routines within the .

### X02 – Machine Constants

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter is concerned with parameters which characterise certain aspects of the computing environment in which the NAG Library is implemented. They relate primarily to floating-point arithmetic, but also to integer arithmetic, the elementary functions and exception handling. The values of the parameters vary from one implementation of the to another, but within the context of a single implementation they are constants.

The parameters are intended for use primarily by other routines in the , but users of the may sometimes need to refer to them directly.

### X03 – Inner Products

 Full details of this chapter for your environment: Fortran Toolbox

This chapter is concerned with the calculation of innerproducts required by other routines within the .

### X04 – Input/Output Utilities

 Full details of this chapter for your environment: Fortran C Toolbox .NET

This chapter contains utility routines concerned with input and output to or from an external file.

### X05 – Date and Time Utilities

 Full details of this chapter for your environment: Fortran Toolbox

This chapter provides routines to obtain the current real time, and the amount of processor time used.

### X06 – OpenMP Utilities

 Full details of this chapter for your environment: Fortran C Toolbox

This chapter contains utilities for controlling the OpenMP environment for your program. They are based on OpenMP runtime library routines, although their functionality varies slightly.

### X07 – IEEE Arithmetic

 Full details of this chapter for your environment: Fortran C

This chapter provides routines to handle various aspects of IEEE floating-point arithmetic behaviour.

### X10 – Automatic Differentiation Utilities

 Full details of this chapter for your environment: Fortran

This chapter is concerned with utility operations used in conjunction with the . Details specific to the can be found in the NAG AD Library Introductuction'.

All routines in the have, as first argument in their interfaces, a handle to a configuration data object which is consistently named throughout the as ad_handle. This argument stores configuration details used in the calculation of algorithmic derivatives. Further details and examples on the use of ad_handle can be found in the NAG AD Library Introductuction'. Chapter X10 provides facilities for:

• Creating or removing a configuration data object
• Setting or getting particular configuration details
• Creating a callback object
• Setting or getting callback object configuration details
• Inserting a callback location into a callback object
• Reading from and writing to a callback object

## Routine Summaries

### A00 – Library Identification

Examples of routines and methods in this chapter:

 a00aa nagf_info_impl_detailsLibrary identification, details of implementation and mark a00ac nagf_info_licenceCheck availability of a valid licence key a00ad nagf_info_impl_details_separateLibrary identification, details of implementation, major and minor marks
Back to A00 Chapter Summary

### A02 – Complex Arithmetic

Examples of routines and methods in this chapter:

 a02aa nagf_complex_sqrtSquare root of complex number a02ab nagf_complex_absModulus of complex number a02ac nagf_complex_divideQuotient of two complex numbers
Back to A02 Chapter Summary

### C02 – Zeros of Polynomials

Examples of routines and methods in this chapter:

 c02af nagf_zeros_poly_complexAll zeros of complex polynomial, modified Laguerre's method c02ag nagf_zeros_poly_realAll zeros of real polynomial, modified Laguerre's method c02ah nagf_zeros_quadratic_complexAll zeros of complex quadratic equation c02aj nagf_zeros_quadratic_realAll zeros of real quadratic equation c02ak nagf_zeros_cubic_realAll zeros of real cubic equation c02al nagf_zeros_quartic_realAll zeros of real quartic equation c02am nagf_zeros_cubic_complexAll zeros of complex cubic equation c02an nagf_zeros_quartic_complexAll zeros of complex quartic equation
Back to C02 Chapter Summary

### C05 – Roots of One or More Transcendental Equations

Examples of routines and methods in this chapter:

 c05au nagf_roots_contfn_brent_intervalZero of continuous function, Brent algorithm, from a given starting value, binary search for interval c05av nagf_roots_contfn_interval_rcommBinary search for interval containing zero of continuous function (reverse communication) c05aw nagf_roots_contfn_cntinZero of continuous function, continuation method, from a given starting value c05ax nagf_roots_contfn_cntin_rcommZero of continuous function, continuation method, from a given starting value (reverse communication) c05ay nagf_roots_contfn_brentZero of continuous function in a given interval, Brent algorithm c05az nagf_roots_contfn_brent_rcommZero of continuous function in a given interval, Brent algorithm (reverse communication) c05ba nagf_roots_lambertw_realReal values of Lambert's W function, W(x) c05bb nagf_roots_lambertw_complexValues of Lambert's W function, W(z) c05md nagf_roots_sys_func_aa_rcommSolution of a system of nonlinear equations using Anderson acceleration (reverse communication) c05qb nagf_roots_sys_func_easySolution of a system of nonlinear equations using function values only (easy-to-use) c05qc nagf_roots_sys_func_expertSolution of a system of nonlinear equations using function values only (comprehensive) c05qd nagf_roots_sys_func_rcommSolution of a system of nonlinear equations using function values only (reverse communication) c05qs nagf_roots_sparsys_func_easySolution of a sparse system of nonlinear equations using function values only (easy-to-use) c05rb nagf_roots_sys_deriv_easySolution of a system of nonlinear equations using first derivatives (easy-to-use) c05rc nagf_roots_sys_deriv_expertSolution of a system of nonlinear equations using first derivatives (comprehensive) c05rd nagf_roots_sys_deriv_rcommSolution of a system of nonlinear equations using first derivatives (reverse communication) c05zd nagf_roots_sys_deriv_checkCheck user's routine for calculating first derivatives of a set of nonlinear functions of several variables
Back to C05 Chapter Summary

### C06 – Summation of Series

Examples of routines and methods in this chapter:

 c06ba nagf_sum_accelerateAcceleration of convergence of sequence, Shanks' transformation and epsilon algorithm c06dc nagf_sum_chebyshevSum of a Chebyshev series at a set of points c06fa nagf_sum_fft_real_1d_rfmtSingle one-dimensional real discrete Fourier transform, extra workspace for greater speed c06fb nagf_sum_fft_hermitian_1d_rfmtSingle one-dimensional Hermitian discrete Fourier transform, extra workspace for greater speed c06fc nagf_sum_fft_complex_1d_sepSingle one-dimensional complex discrete Fourier transform, extra workspace for greater speed c06ff nagf_sum_fft_complex_multid_1d_sepOne-dimensional complex discrete Fourier transform of multidimensional data c06fj nagf_sum_fft_complex_multid_sepMultidimensional complex discrete Fourier transform of multidimensional data c06fk nagf_sum_convcorr_realCircular convolution or correlation of two real vectors, no restrictions on n c06fx nagf_sum_fft_complex_3d_sepThree-dimensional complex discrete Fourier transform c06la nagf_sum_invlaplace_crumpInverse Laplace transform, Crump's method c06lb nagf_sum_invlaplace_weeksInverse Laplace transform, modified Weeks' method c06lc nagf_sum_invlaplace_weeks_evalEvaluate inverse Laplace transform as computed by c06lb c06pa nagf_sum_fft_realherm_1dSingle one-dimensional real and Hermitian complex discrete Fourier transform, using complex storage format for Hermitian sequences c06pc nagf_sum_fft_complex_1dSingle one-dimensional complex discrete Fourier transform, complex data type c06pf nagf_sum_fft_complex_multid_1dOne-dimensional complex discrete Fourier transform of multidimensional data (using complex data type) c06pj nagf_sum_fft_complex_multidMultidimensional complex discrete Fourier transform of multidimensional data (using complex data type) c06pk nagf_sum_convcorr_complexCircular convolution or correlation of two complex vectors c06pp nagf_sum_fft_realherm_1d_multi_rowMultiple one-dimensional real and Hermitian complex discrete Fourier transforms, using row ordered complex storage format for Hermitian sequences c06pq nagf_sum_fft_realherm_1d_multi_colMultiple one-dimensional real and Hermitian complex discrete Fourier transforms, using column ordered complex storage format for Hermitian sequences c06pr nagf_sum_fft_complex_1d_multi_rowMultiple one-dimensional complex discrete Fourier transforms using complex data type c06ps nagf_sum_fft_complex_1d_multi_colMultiple one-dimensional complex discrete Fourier transforms, complex data type c06pu nagf_sum_fft_complex_2dTwo-dimensional complex discrete Fourier transform, complex data type c06pv nagf_sum_fft_real_2dTwo-dimensional real-to-complex discrete Fourier transform c06pw nagf_sum_fft_hermitian_2dTwo-dimensional complex-to-real discrete Fourier transform c06px nagf_sum_fft_complex_3dThree-dimensional complex discrete Fourier transform, complex data type c06py nagf_sum_fft_real_3dThree-dimensional real-to-complex discrete Fourier transform c06pz nagf_sum_fft_hermitian_3dThree-dimensional complex-to-real discrete Fourier transform c06ra nagf_sum_fft_real_sine_simpleDiscrete sine transform (easy-to-use) c06rb nagf_sum_fft_real_cosine_simpleDiscrete cosine transform (easy-to-use) c06rc nagf_sum_fft_real_qtrsine_simpleDiscrete quarter-wave sine transform (easy-to-use) c06rd nagf_sum_fft_real_qtrcosine_simpleDiscrete quarter-wave cosine transform (easy-to-use) c06re nagf_sum_fft_sineMultiple discrete sine transforms, simple c06rf nagf_sum_fft_cosineMultiple discrete cosine transforms, simple c06rg nagf_sum_fft_qtrsineMultiple discrete quarter-wave sine transforms, simple c06rh nagf_sum_fft_qtrcosineMultiple discrete quarter-wave cosine transforms, simple c06sa nagf_sum_fast_gaussMultidimensional fast Gauss transform
Back to C06 Chapter Summary

### C09 – Wavelet Transforms

Examples of routines and methods in this chapter:

 c09aa nagf_wav_1d_initOne-dimensional wavelet filter initialization c09ab nagf_wav_2d_initTwo-dimensional wavelet filter initialization c09ac nagf_wav_3d_initThree-dimensional wavelet filter initialization c09ba nagf_wav_1d_contOne-dimensional real continuous wavelet transform c09ca nagf_wav_1d_sngl_fwdOne-dimensional discrete wavelet transform c09cb nagf_wav_1d_sngl_invOne-dimensional inverse discrete wavelet transform c09cc nagf_wav_1d_multi_fwdOne-dimensional multi-level discrete wavelet transform c09cd nagf_wav_1d_multi_invOne-dimensional inverse multi-level discrete wavelet transform c09da nagf_wav_1d_mxolap_fwdOne-dimensional maximal overlap discrete wavelet transform (MODWT) c09db nagf_wav_1d_mxolap_invOne-dimensional inverse maximal overlap discrete wavelet transform (IMODWT) c09dc nagf_wav_1d_mxolap_multi_fwdOne-dimensional multi-level maximal overlap discrete wavelet transform (MODWT) c09dd nagf_wav_1d_mxolap_multi_invOne-dimensional inverse multi-level maximal overlap discrete wavelet transform (IMODWT) c09ea nagf_wav_2d_sngl_fwdTwo-dimensional discrete wavelet transform c09eb nagf_wav_2d_sngl_invTwo-dimensional inverse discrete wavelet transform c09ec nagf_wav_2d_multi_fwdTwo-dimensional multi-level discrete wavelet transform c09ed nagf_wav_2d_multi_invTwo-dimensional inverse multi-level discrete wavelet transform c09ey nagf_wav_2d_coeff_extTwo-dimensional discrete wavelet transform coefficient extraction c09ez nagf_wav_2d_coeff_insTwo-dimensional discrete wavelet transform coefficient insertion c09fa nagf_wav_3d_sngl_fwdThree-dimensional discrete wavelet transform c09fb nagf_wav_3d_sngl_invThree-dimensional inverse discrete wavelet transform c09fc nagf_wav_3d_multi_fwdThree-dimensional multi-level discrete wavelet transform c09fd nagf_wav_3d_mxolap_multi_invThree-dimensional inverse multi-level discrete wavelet transform c09fy nagf_wav_3d_coeff_extThree-dimensional discrete wavelet transform coefficient extraction c09fz nagf_wav_3d_coeff_insThree-dimensional discrete wavelet transform coefficient insertion
Back to C09 Chapter Summary

Examples of routines and methods in this chapter:

Back to D01 Chapter Summary

### D02 – Ordinary Differential Equations Integrators for Stiff Ordinary Differential Systems

Examples of routines and methods in this chapter:

Back to D02 Chapter Summary

### D03 – Partial Differential Equations

Examples of routines and methods in this chapter:

 d03ea nagf_pde_2d_laplaceElliptic PDE, Laplace's equation, two-dimensional arbitrary domain d03eb nagf_pde_2d_ellip_fdElliptic PDE, solution of finite difference equations by SIP, five-point two-dimensional molecule, iterate to convergence d03ec nagf_pde_3d_ellip_fdElliptic PDE, solution of finite difference equations by SIP for seven-point three-dimensional molecule, iterate to convergence d03ed nagf_pde_2d_ellip_mgridElliptic PDE, solution of finite difference equations by a multigrid technique d03ee nagf_pde_2d_ellip_discretDiscretize a second-order elliptic PDE on a rectangle d03fa nagf_pde_3d_ellip_helmholtzElliptic PDE, Helmholtz equation, three-dimensional Cartesian coordinates d03ma nagf_pde_2d_triangulateTriangulation of plane region d03nc nagf_pde_1d_blackscholes_fdFinite difference solution of the Black–Scholes equations d03nd nagf_pde_1d_blackscholes_closedAnalytic solution of the Black–Scholes equations d03ne nagf_pde_1d_blackscholes_meansCompute average values for d03nd d03pc nagf_pde_1d_parab_fd_oldGeneral system of parabolic PDEs, method of lines, finite differences, one space variable d03pc nagf_pde_1d_parab_remesh_fd_dummy_odedef_olddummy d03pc nagf_pde_1d_parab_remesh_fd_dummy_monitf_olddummy d03pd nagf_pde_1d_parab_coll_oldGeneral system of parabolic PDEs, method of lines, Chebyshev C0 collocation, one space variable d03pe nagf_pde_1d_parab_kellerGeneral system of first-order PDEs, method of lines, Keller box discretization, one space variable d03pe nagf_pde_1d_parab_dae_keller_remesh_fd_dummy_odedefdummy d03pe nagf_pde_1d_parab_dae_keller_remesh_fd_dummy_monitfdummy d03pf nagf_pde_1d_parab_convdiffGeneral system of convection-diffusion PDEs with source terms in conservative form, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable d03ph nagf_pde_1d_parab_dae_fd_oldGeneral system of parabolic PDEs, coupled DAEs, method of lines, finite differences, one space variable d03pj nagf_pde_1d_parab_dae_coll_oldGeneral system of parabolic PDEs, coupled DAEs, method of lines, Chebyshev C0 collocation, one space variable d03pk nagf_pde_1d_parab_dae_kellerGeneral system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, one space variable d03pl nagf_pde_1d_parab_convdiff_daeGeneral system of convection-diffusion PDEs with source terms in conservative form, coupled DAEs, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable d03pp nagf_pde_1d_parab_remesh_fd_oldGeneral system of parabolic PDEs, coupled DAEs, method of lines, finite differences, remeshing, one space variable d03pr nagf_pde_1d_parab_remesh_kellerGeneral system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, remeshing, one space variable d03ps nagf_pde_1d_parab_convdiff_remeshGeneral system of convection-diffusion PDEs, coupled DAEs, method of lines, upwind scheme, remeshing, one space variable d03pu nagf_pde_1d_parab_euler_roeRoe's approximate Riemann solver for Euler equations in conservative form, for use with d03pf, d03pl and d03ps d03pv nagf_pde_1d_parab_euler_osherOsher's approximate Riemann solver for Euler equations in conservative form, for use with d03pf, d03pl and d03ps d03pw nagf_pde_1d_parab_euler_hllModified HLL Riemann solver for Euler equations in conservative form, for use with d03pf, d03pl and d03ps d03px nagf_pde_1d_parab_euler_exactExact Riemann solver for Euler equations in conservative form, for use with d03pf, d03pl and d03ps d03py nagf_pde_1d_parab_coll_interpPDEs, spatial interpolation with d03pd or d03pj d03pz nagf_pde_1d_parab_fd_interpPDEs, spatial interpolation with d03pc, d03pe, d03pf, d03ph, d03pk, d03pl, d03pp, d03pr or d03ps d03ra nagf_pde_2d_gen_order2_rectangleGeneral system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectangular region d03rb nagf_pde_2d_gen_order2_rectilinearGeneral system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectilinear region d03rz nagf_pde_2d_gen_order2_rectilinear_extractgridExtract grid data from d03rb d03ua nagf_pde_2d_ellip_fd_iterElliptic PDE, solution of finite difference equations by SIP, five-point two-dimensional molecule, one iteration d03ub nagf_pde_3d_ellip_fd_iterElliptic PDE, solution of finite difference equations by SIP, seven-point three-dimensional molecule, one iteration
Back to D03 Chapter Summary

### D04 – Numerical Differentiation

Examples of routines and methods in this chapter:

 d04aa nagf_numdiff_fwdNumerical differentiation, derivatives up to order 14, function of one real variable d04ba nagf_numdiff_rcommNumerical differentiation, user-supplied function values, derivatives up to order 14, derivatives with respect to one real variable d04bb nagf_numdiff_sampleGenerates sample points for function evaluations by d04ba
Back to D04 Chapter Summary

### D05 – Integral Equations

Examples of routines and methods in this chapter:

 d05aa nagf_inteq_fredholm2_splitLinear nonsingular Fredholm integral equation, second kind, split kernel d05ab nagf_inteq_fredholm2_smoothLinear nonsingular Fredholm integral equation, second kind, smooth kernel d05ba nagf_inteq_volterra2Nonlinear Volterra convolution equation, second kind d05bd nagf_inteq_abel2_weakNonlinear convolution Volterra–Abel equation, second kind, weakly singular d05be nagf_inteq_abel1_weakNonlinear convolution Volterra–Abel equation, first kind, weakly singular d05bw nagf_inteq_volterra_weightsGenerate weights for use in solving Volterra equations d05by nagf_inteq_abel_weak_weightsGenerate weights for use in solving weakly singular Abel-type equations
Back to D05 Chapter Summary

### D06 – Mesh Generation

Examples of routines and methods in this chapter:

 d06aa nagf_mesh_2d_gen_incGenerates a two-dimensional mesh using a simple incremental method d06ab nagf_mesh_2d_gen_delaunayGenerates a two-dimensional mesh using a Delaunay–Voronoi process d06ac nagf_mesh_2d_gen_frontGenerates a two-dimensional mesh using an Advancing-front method d06ba nagf_mesh_2d_gen_boundaryGenerates a boundary mesh d06ca nagf_mesh_2d_smooth_baryUses a barycentering technique to smooth a given mesh d06cb nagf_mesh_2d_sparsityGenerates a sparsity pattern of a Finite Element matrix associated with a given mesh d06cc nagf_mesh_2d_renumberRenumbers a given mesh using Gibbs method d06da nagf_mesh_2d_transform_affineGenerates a mesh resulting from an affine transformation of a given mesh d06db nagf_mesh_2d_joinJoins together two given adjacent (possibly overlapping) meshes
Back to D06 Chapter Summary

### E01 – Interpolation

Examples of routines and methods in this chapter:

 e01aa nagf_interp_dim1_aitkenInterpolated values, Aitken's technique, unequally spaced data, one variable e01ab nagf_interp_dim1_everettInterpolated values, Everett's formula, equally spaced data, one variable e01ae nagf_interp_dim1_chebInterpolating functions, polynomial interpolant, data may include derivative values, one variable e01ba nagf_interp_dim1_splineInterpolating functions, cubic spline interpolant, one variable e01be nagf_interp_dim1_monotonicInterpolating functions, monotonicity-preserving, piecewise cubic Hermite, one variable e01bf nagf_interp_dim1_monotonic_evalInterpolated values, interpolant computed by e01be, function only, one variable e01bg nagf_interp_dim1_monotonic_derivInterpolated values, interpolant computed by e01be, function and first derivative, one variable e01bh nagf_interp_dim1_monotonic_intgInterpolated values, interpolant computed by e01be, definite integral, one variable e01da nagf_interp_2d_spline_gridInterpolating functions, fitting bicubic spline, data on rectangular grid e01ea nagf_interp_2d_triangulateTriangulation of two-dimensional scattered grid, method of Renka and Cline e01eb nagf_interp_2d_triang_bary_evalBarycentric interpolation on function values provided on a two-dimensional scattered grid e01ra nagf_interp_dim1_ratnlInterpolating functions, rational interpolant, one variable e01rb nagf_interp_dim1_ratnl_evalInterpolated values, evaluate rational interpolant computed by e01ra, one variable e01sa nagf_interp_2d_scatInterpolating functions, method of Renka and Cline, two variables e01sb nagf_interp_2d_scat_evalInterpolated values, evaluate interpolant computed by e01sa, two variables e01sg nagf_interp_2d_scat_shepInterpolating functions, modified Shepard's method, two variables e01sh nagf_interp_2d_scat_shep_evalInterpolated values, evaluate interpolant computed by e01sg, function and first derivatives, two variables e01tg nagf_interp_3d_scat_shepInterpolating functions, modified Shepard's method, three variables e01th nagf_interp_3d_scat_shep_evalInterpolated values, evaluate interpolant computed by e01tg, function and first derivatives, three variables e01tk nagf_interp_4d_scat_shepInterpolating functions, modified Shepard's method, four variables e01tl nagf_interp_4d_scat_shep_evalInterpolated values, evaluate interpolant computed by e01tk, function and first derivatives, four variables e01tm nagf_interp_5d_scat_shepInterpolating functions, modified Shepard's method, five variables e01tn nagf_interp_5d_scat_shep_evalInterpolated values, evaluate interpolant computed by e01tm, function and first derivatives, five variables e01zm nagf_interp_nd_scat_shepInterpolating function, modified Shepard's method, d dimensions e01zn nagf_interp_nd_scat_shep_evalInterpolated values, evaluate interpolant computed by e01zm, function and first derivatives, d dimensions
Back to E01 Chapter Summary

### E02 – Curve and Surface Fitting

Examples of routines and methods in this chapter:

 e02ad nagf_fit_1dcheb_arbLeast squares curve fit, by polynomials, arbitrary data points e02ae nagf_fit_1dcheb_evalEvaluation of fitted polynomial in one variable from Chebyshev series form (simplified parameter list) e02af nagf_fit_1dcheb_glpLeast squares polynomial fit, special data points (including interpolation) e02ag nagf_fit_1dcheb_conLeast squares polynomial fit, values and derivatives may be constrained, arbitrary data points e02ah nagf_fit_1dcheb_derivDerivative of fitted polynomial in Chebyshev series form e02aj nagf_fit_1dcheb_integIntegral of fitted polynomial in Chebyshev series form e02ak nagf_fit_1dcheb_eval2Evaluation of fitted polynomial in one variable from Chebyshev series form e02al nagf_fit_1d_minimax_polynomialMinimax curve fit by polynomials e02ba nagf_fit_1dspline_knotsLeast squares curve cubic spline fit (including interpolation) e02bb nagf_fit_1dspline_evalEvaluation of fitted cubic spline, function only e02bc nagf_fit_1dspline_derivEvaluation of fitted cubic spline, function and derivatives e02bd nagf_fit_1dspline_integEvaluation of fitted cubic spline, definite integral e02be nagf_fit_1dspline_autoLeast squares cubic spline curve fit, automatic knot placement e02bf nagf_fit_1dspline_deriv_vectorEvaluation of fitted cubic spline, function and optionally derivatives at a vector of points e02ca nagf_fit_2dcheb_linesLeast squares surface fit by polynomials, data on lines parallel to one independent coordinate axis e02cb nagf_fit_2dcheb_evalEvaluation of fitted polynomial in two variables e02da nagf_fit_2dspline_panelLeast squares surface fit, bicubic splines e02dc nagf_fit_2dspline_gridLeast squares surface fit by bicubic splines with automatic knot placement, data on rectangular grid e02dd nagf_fit_2dspline_sctrLeast squares surface fit by bicubic splines with automatic knot placement, scattered data e02de nagf_fit_2dspline_evalvEvaluation of fitted bicubic spline at a vector of points e02df nagf_fit_2dspline_evalmEvaluation of fitted bicubic spline at a mesh of points e02dh nagf_fit_2dspline_derivmEvaluation of spline surface at mesh of points with derivatives e02ga nagf_fit_glin_l1solL1-approximation by general linear function e02gb nagf_fit_glinc_l1solL1-approximation by general linear function subject to linear inequality constraints e02gc nagf_fit_glin_linfL∞-approximation by general linear function e02jd nagf_fit_2dspline_ts_sctrSpline approximation to a set of scattered data using a two-stage approximation method e02je nagf_fit_2dspline_ts_evalvEvaluation at a vector of points of a spline computed by e02jd e02jf nagf_fit_2dspline_ts_evalmEvaluation at a mesh of points of a spline computed by e02jd e02ra nagf_fit_pade_appPadé approximants e02rb nagf_fit_pade_evalEvaluation of fitted rational function as computed by e02ra e02za nagf_fit_2dspline_sortSort two-dimensional data into panels for fitting bicubic splines e02zk nagf_fit_opt_setOption setting routine e02zl nagf_fit_opt_getOption getting routine
Back to E02 Chapter Summary

### E04 – Minimizing or Maximizing a Function

Examples of routines and methods in this chapter:

Back to E04 Chapter Summary

### E05 – Global Optimization of a Function

Examples of routines and methods in this chapter:

 e05ja nagf_glopt_bnd_mcs_initInitialization routine for e05jb e05jb nagf_glopt_bnd_mcs_solveGlobal optimization by multi-level coordinate search, simple bounds, using function values only e05jc nagf_glopt_bnd_mcs_optset_fileSupply optional parameter values for e05jb from external file e05jd nagf_glopt_bnd_mcs_optset_stringSet a single optional parameter for e05jb from a character string e05je nagf_glopt_bnd_mcs_optset_charSet a single optional parameter for e05jb from an 'ON'/'OFF'-valued character argument e05jf nagf_glopt_bnd_mcs_optset_intSet a single optional parameter for e05jb from an integer argument e05jg nagf_glopt_bnd_mcs_optset_realSet a single optional parameter for e05jb from a real argument e05jh nagf_glopt_bnd_mcs_option_checkDetermine whether an optional parameter for e05jb has been set by you or not e05jj nagf_glopt_bnd_mcs_optget_charGet the setting of an 'ON'/'OFF'-valued character optional parameter of e05jb e05jk nagf_glopt_bnd_mcs_optget_intGet the setting of an integer valued optional parameter of e05jb e05jl nagf_glopt_bnd_mcs_optget_realGet the setting of a real valued optional parameter of e05jb e05sa nagf_glopt_bnd_psoGlobal optimization using particle swarm algorithm (PSO), bound constraints only e05sb nagf_glopt_nlp_psoGlobal optimization using particle swarm algorithm (PSO), comprehensive e05sx nagf_glopt_bnd_pso_dummy_monmoddummy e05sy nagf_glopt_nlp_pso_dummy_monmoddummy e05uc nagf_glopt_nlp_multistart_sqpGlobal optimization using multi-start, nonlinear constraints e05ud nagf_glopt_nlp_multistart_dcfdummy e05us nagf_glopt_nlp_multistart_sqp_lsqGlobal optimization of a sum of squares problem using multi-start, nonlinear constraints e05zk nagf_glopt_optsetOption setting routine for e05sa, e05sb, e05uc and e05us e05zl nagf_glopt_optgetOption getting routine for e05sa, e05sb, e05uc and e05us
Back to E05 Chapter Summary

### F01 – Matrix Operations, Including Inversion

Examples of routines and methods in this chapter:

 f01ab nagf_matop_real_symm_posdef_invInverse of real symmetric positive definite matrix using iterative refinement f01ad nagf_matop_real_symm_posdef_inv_norefInverse of real symmetric positive definite matrix f01bl nagf_matop_real_gen_pseudinvPseudo-inverse and rank of real m by n matrix (m≥n) f01br nagf_matop_real_gen_sparse_luLU factorization of real sparse matrix f01bs nagf_matop_real_gen_sparse_lu_reuseLU factorization of real sparse matrix with known sparsity pattern f01bu nagf_matop_real_symm_posdef_facULDLTUT factorization of real symmetric positive definite band matrix f01bv nagf_matop_real_symm_posdef_geneigReduction to standard form, generalized real symmetric-definite banded eigenproblem f01ck nagf_matop_real_gen_matmulMultiplication of real matrices f01cr nagf_matop_real_gen_trans_inplaceTransposition of a real matrix f01ct nagf_matop_real_addsubSum or difference of two real matrices, optional scaling and transposition f01cw nagf_matop_complex_addsubSum or difference of two complex matrices, optional scaling and transposition f01ec nagf_matop_real_gen_matrix_expReal matrix exponential f01ed nagf_matop_real_symm_matrix_expReal symmetric matrix exponential f01ef nagf_matop_real_symm_matrix_funFunction of a real symmetric matrix f01ej nagf_matop_real_gen_matrix_logReal matrix logarithm f01ek nagf_matop_real_gen_matrix_fun_stdExponential, sine, cosine, sinh or cosh of a real matrix (Schur–Parlett algorithm) f01el nagf_matop_real_gen_matrix_fun_numFunction of a real matrix (using numerical differentiation) f01em nagf_matop_real_gen_matrix_fun_usdFunction of a real matrix (using user-supplied derivatives) f01en nagf_matop_real_gen_matrix_sqrtReal matrix square root f01ep nagf_matop_real_tri_matrix_sqrtReal upper quasi-triangular matrix square root f01eq nagf_matop_real_gen_matrix_powGeneral power of a real matrix f01fc nagf_matop_complex_gen_matrix_expComplex matrix exponential f01fd nagf_matop_complex_herm_matrix_expComplex Hermitian matrix exponential f01ff nagf_matop_complex_herm_matrix_funFunction of a complex Hermitian matrix f01fj nagf_matop_complex_gen_matrix_logComplex matrix logarithm f01fk nagf_matop_complex_gen_matrix_fun_stdExponential, sine, cosine, sinh or cosh of a complex matrix (Schur–Parlett algorithm) f01fl nagf_matop_complex_gen_matrix_fun_numFunction of a complex matrix (using numerical differentiation) f01fm nagf_matop_complex_gen_matrix_fun_usdFunction of a complex matrix (using user-supplied derivatives) f01fn nagf_matop_complex_gen_matrix_sqrtComplex matrix square root f01fp nagf_matop_complex_tri_matrix_sqrtComplex upper triangular matrix square root f01fq nagf_matop_complex_gen_matrix_powGeneral power of a complex matrix f01ga nagf_matop_real_gen_matrix_actexpAction of a real matrix exponential on a real matrix f01gb nagf_matop_real_gen_matrix_actexp_rcommAction of a real matrix exponential on a real matrix (reverse communication) f01ha nagf_matop_complex_gen_matrix_actexpAction of a complex matrix exponential on a complex matrix f01hb nagf_matop_complex_gen_matrix_actexp_rcommAction of a complex matrix exponential on a complex matrix (reverse communication) f01ja nagf_matop_real_gen_matrix_cond_stdCondition number for the exponential, logarithm, sine, cosine, sinh or cosh of a real matrix f01jb nagf_matop_real_gen_matrix_cond_numCondition number for a function of a real matrix (using numerical differentiation) f01jc nagf_matop_real_gen_matrix_cond_usdCondition number for a function of a real matrix (using user-supplied derivatives) f01jd nagf_matop_real_gen_matrix_cond_sqrtCondition number for square root of real matrix f01je nagf_matop_real_gen_matrix_cond_powCondition number for real matrix power f01jf nagf_matop_real_gen_matrix_frcht_powFréchet derivative of real matrix power f01jg nagf_matop_real_gen_matrix_cond_expCondition number for real matrix exponential f01jh nagf_matop_real_gen_matrix_frcht_expFréchet derivative of real matrix exponential f01jj nagf_matop_real_gen_matrix_cond_logCondition number for real matrix logarithm f01jk nagf_matop_real_gen_matrix_frcht_logFréchet derivative of real matrix logarithm f01ka nagf_matop_complex_gen_matrix_cond_stdCondition number for the exponential, logarithm, sine, cosine, sinh or cosh of a complex matrix f01kb nagf_matop_complex_gen_matrix_cond_numCondition number for a function of a complex matrix (using numerical differentiation) f01kc nagf_matop_complex_gen_matrix_cond_usdCondition number for a function of a complex matrix (using user-supplied derivatives) f01kd nagf_matop_complex_gen_matrix_cond_sqrtCondition number for square root of complex matrix f01ke nagf_matop_complex_gen_matrix_cond_powCondition number for complex matrix power f01kf nagf_matop_complex_gen_matrix_frcht_powFréchet derivative of complex matrix power f01kg nagf_matop_complex_gen_matrix_cond_expCondition number for complex matrix exponential f01kh nagf_matop_complex_gen_matrix_frcht_expFréchet derivative of complex matrix exponential f01kj nagf_matop_complex_gen_matrix_cond_logCondition number for complex matrix logarithm f01kk nagf_matop_complex_gen_matrix_frcht_logFréchet derivative of complex matrix logarithm f01le nagf_matop_real_gen_tridiag_luLU factorization of real tridiagonal matrix f01lh nagf_matop_real_gen_blkdiag_luLU factorization of real almost block diagonal matrix f01mc nagf_matop_real_vband_posdef_facLDLT factorization of real symmetric positive definite variable-bandwidth matrix f01qg nagf_matop_real_trapez_rqRQ factorization of real m by n upper trapezoidal matrix (m≤n) f01qj nagf_matop_real_gen_rqRQ factorization of real m by n matrix (m≤n) f01qk nagf_matop_real_gen_rq_formqOperations with orthogonal matrices, form rows of Q, after RQ factorization by f01qj f01rg nagf_matop_complex_trapez_rqRQ factorization of complex m by n upper trapezoidal matrix (m≤n) f01rj nagf_matop_complex_gen_rqRQ factorization of complex m by n matrix (m≤n) f01rk nagf_matop_complex_gen_rq_formqOperations with unitary matrices, form rows of Q, after RQ factorization by f01rj f01va nagf_matop_dtrttpCopies a real triangular matrix from full format to packed format f01vb nagf_matop_ztrttpCopies a complex triangular matrix from full format to packed format f01vc nagf_matop_dtpttrCopies a real triangular matrix from packed format to full format f01vd nagf_matop_ztpttrCopies a complex triangular matrix from packed format to full format f01ve nagf_matop_dtrttfCopies a real triangular matrix from full format to Rectangular Full Packed format f01vf nagf_matop_ztrttfCopies a complex triangular matrix from full format to Rectangular Full Packed format f01vg nagf_matop_dtfttrCopies a real triangular matrix from Rectangular Full Packed format to full format f01vh nagf_matop_ztfttrCopies a complex triangular matrix from Rectangular Full Packed format to full format f01vj nagf_matop_dtpttfCopies a real triangular matrix from packed format to Rectangular Full Packed format f01vk nagf_matop_ztpttfCopies a complex triangular matrix from packed format to Rectangular Full Packed format f01vl nagf_matop_dtfttpCopies a real triangular matrix from Rectangular Full Packed format to packed format f01vm nagf_matop_ztfttpCopies a complex triangular matrix from Rectangular Full Packed format to packed format f01za nagf_matop_real_tri_packConvert real matrix between packed triangular and square storage formats f01zb nagf_matop_complex_tri_packConvert complex matrix between packed triangular and square storage formats f01zc nagf_matop_real_band_packConvert real matrix between packed banded and rectangular storage formats f01zd nagf_matop_complex_band_packConvert complex matrix between packed banded and rectangular storage formats
Back to F01 Chapter Summary

### F02 – Eigenvalues and Eigenvectors

Examples of routines and methods in this chapter:

 f02ec nagf_eigen_real_gen_eigsysSelected eigenvalues and eigenvectors of real nonsymmetric matrix (Black Box) f02ek nagf_eigen_real_gen_sparse_arnoldiSelected eigenvalues and eigenvectors of a real sparse general matrix f02ek nagf_eigen_arnoldi_optiondummy f02ek nagf_eigen_arnoldi_monit_gendummy f02fj nagf_eigen_real_symm_sparse_eigsysSelected eigenvalues and eigenvectors of sparse symmetric eigenproblem (Black Box) f02fj nagf_eigen_monitdummy f02fk nagf_eigen_real_symm_sparse_arnoldiSelected eigenvalues and eigenvectors of a real symmetric sparse matrix f02fk nagf_eigen_arnoldi_monit_symmdummy f02gc nagf_eigen_complex_gen_eigsysSelected eigenvalues and eigenvectors of complex nonsymmetric matrix (Black Box) f02jc nagf_eigen_real_gen_quadSolves the quadratic eigenvalue problem for real matrices f02jq nagf_eigen_complex_gen_quadSolves the quadratic eigenvalue problem for complex matrices f02wg nagf_eigen_real_gen_partialsvdComputes leading terms in the singular value decomposition of a real general matrix; also computes corresponding left and right singular vectors f02wu nagf_eigen_real_triang_svdSVD of real upper triangular matrix (Black Box) f02xu nagf_eigen_complex_triang_svdSVD of complex upper triangular matrix (Black Box)
Back to F02 Chapter Summary

### F03 – Determinants

Examples of routines and methods in this chapter:

 f03ba nagf_det_real_genDeterminant of real matrix, matrix already factorized by f07ad f03bf nagf_det_real_symDeterminant of real symmetric positive definite matrix f03bh nagf_det_real_band_symDeterminant of real symmetric positive definite banded matrix previously factorized by f07hd f03bn nagf_det_complex_genDeterminant of complex matrix previously LU factorized
Back to F03 Chapter Summary

### F04 – Simultaneous Linear Equations

Examples of routines and methods in this chapter:

 f04am nagf_linsys_real_gen_lsqsolLeast squares solution of m real equations in n unknowns, rank =n, m≥n using iterative refinement (Black Box) f04ax nagf_linsys_real_sparse_fac_solveSolution of real sparse simultaneous linear equations (coefficient matrix already factorized) f04ba nagf_linsys_real_square_solveComputes the solution, estimated condition number and error-bound to a real system of linear equations f04bb nagf_linsys_real_band_solveComputes the solution, estimated condition number and error-bound to a real banded system of linear equations f04bc nagf_linsys_real_tridiag_solveComputes the solution, estimated condition number and error-bound to a real tridiagonal system of linear equations f04bd nagf_linsys_real_posdef_solveComputes the solution, estimated condition number and error-bound to a real symmetric positive definite system of linear equations f04be nagf_linsys_real_posdef_packed_solveComputes the solution, estimated condition number and error-bound to a real symmetric positive definite system of linear equations, packed storage f04bf nagf_linsys_real_posdef_band_solveComputes the solution, estimated condition number and error-bound to a real symmetric positive definite banded system of linear equations f04bg nagf_linsys_real_posdef_tridiag_solveComputes the solution, estimated condition number and error-bound to a real symmetric positive definite tridiagonal system of linear equations f04bh nagf_linsys_real_symm_solveComputes the solution, estimated condition number and error-bound to a real symmetric system of linear equations f04bj nagf_linsys_real_symm_packed_solveComputes the solution, estimated condition number and error-bound to a real symmetric system of linear equations, packed storage f04ca nagf_linsys_complex_square_solveComputes the solution, estimated condition number and error-bound to a complex system of linear equations f04cb nagf_linsys_complex_band_solveComputes the solution, estimated condition number and error-bound to a complex banded system of linear equations f04cc nagf_linsys_complex_tridiag_solveComputes the solution, estimated condition number and error-bound to a complex tridiagonal system of linear equations f04cd nagf_linsys_complex_posdef_solveComputes the solution, estimated condition number and error-bound to a complex Hermitian positive definite system of linear equations f04ce nagf_linsys_complex_posdef_packed_solveComputes the solution, estimated condition number and error-bound to a complex Hermitian positive definite system of linear equations, packed storage f04cf nagf_linsys_complex_posdef_band_solveComputes the solution, estimated condition number and error-bound to a complex Hermitian positive definite banded system of linear equations f04cg nagf_linsys_complex_posdef_tridiag_solveComputes the solution, estimated condition number and error-bound to a complex Hermitian positive definite tridiagonal system of linear equations f04ch nagf_linsys_complex_herm_solveComputes the solution and error-bound to a complex Hermitian system of linear equations f04cj nagf_linsys_complex_herm_packed_solveComputes the solution, estimated condition number and error-bound to a complex Hermitian system of linear equations, packed storage f04dh nagf_linsys_complex_symm_solveComputes the solution, estimated condition number and error-bound to a complex symmetric system of linear equations f04dj nagf_linsys_complex_symm_packed_solveComputes the solution, estimated condition number and error-bound to a complex symmetric system of linear equations, packed storage f04fe nagf_linsys_real_toeplitz_yuleSolution of the Yule–Walker equations for real symmetric positive definite Toeplitz matrix, one right-hand side f04ff nagf_linsys_real_toeplitz_solveSolution of real symmetric positive definite Toeplitz system, one right-hand side f04jg nagf_linsys_real_gen_solveLeast squares (if rank =n) or minimal least squares (if rank
Back to F04 Chapter Summary

### F05 – Orthogonalization

Examples of routines and methods in this chapter:

 f05aa nagf_orthog_real_gram_schmidtGram–Schmidt orthogonalization of n vectors of order m
Back to F05 Chapter Summary

### F06 – Linear Algebra Support Routines

Examples of routines and methods in this chapter:

Back to F06 Chapter Summary

### F07 – Linear Equations (LAPACK)

Examples of routines and methods in this chapter:

Back to F07 Chapter Summary

### F08 – Least Squares and Eigenvalue Problems (LAPACK)

Examples of routines and methods in this chapter:

 f08aa nagf_lapackeig_dgelsSolves a real linear least squares problem of full rank f08ab nagf_lapackeig_dgeqrtPerforms a QR factorization of real general rectangular matrix, with explicit blocking f08ac nagf_lapackeig_dgemqrtApplies the orthogonal transformation determined by f08ab f08ae nagf_lapackeig_dgeqrfPerforms a QR factorization of real general rectangular matrix f08af nagf_lapackeig_dorgqrForms all or part of orthogonal Q from QR factorization determined by f08ae, f08be and f08bf f08ag nagf_lapackeig_dormqrApplies an orthogonal transformation determined by f08ae, f08be and f08bf f08ah nagf_lapackeig_dgelqfPerforms a LQ factorization of real general rectangular matrix f08aj nagf_lapackeig_dorglqForms all or part of orthogonal Q from LQ factorization determined by f08ah f08ak nagf_lapackeig_dormlqApplies the orthogonal transformation determined by f08ah f08an nagf_lapackeig_zgelsSolves a complex linear least problem of full rank f08ap nagf_lapackeig_zgeqrtPerforms a QR factorization of complex general rectangular matrix using recursive algorithm f08aq nagf_lapackeig_zgemqrtApplies the unitary transformation determined by f08ap f08as nagf_lapackeig_zgeqrfPerforms a QR factorization of complex general rectangular matrix f08at nagf_lapackeig_zungqrForms all or part of unitary Q from QR factorization determined by f08as, f08bs and f08bt f08au nagf_lapackeig_zunmqrApplies a unitary transformation determined by f08as, f08bs and f08bt f08av nagf_lapackeig_zgelqfPerforms a LQ factorization of complex general rectangular matrix f08aw nagf_lapackeig_zunglqForms all or part of unitary Q from LQ factorization determined by f08av f08ax nagf_lapackeig_zunmlqApplies the unitary transformation determined by f08av f08ba nagf_lapackeig_dgelsyComputes the minimum-norm solution to a real linear least squares problem f08bb nagf_lapackeig_dtpqrtQR factorization of real general triangular-pentagonal matrix f08bc nagf_lapackeig_dtpmqrtApplies the orthogonal transformation determined by f08bb f08be nagf_lapackeig_dgeqpfQR factorization, with column pivoting, of real general rectangular matrix f08bf nagf_lapackeig_dgeqp3QR factorization, with column pivoting, using BLAS-3, of real general rectangular matrix f08bh nagf_lapackeig_dtzrzfReduces a real upper trapezoidal matrix to upper triangular form f08bk nagf_lapackeig_dormrzApplies the orthogonal transformation determined by f08bh f08bn nagf_lapackeig_zgelsyComputes the minimum-norm solution to a complex linear least squares problem f08bp nagf_lapackeig_ztpqrtQR factorization of complex triangular-pentagonal matrix f08bq nagf_lapackeig_ztpmqrtApplies the unitary transformation determined by f08bp f08bs nagf_lapackeig_zgeqpfQR factorization, with column pivoting, of complex general rectangular matrix f08bt nagf_lapackeig_zgeqp3QR factorization, with column pivoting, using BLAS-3, of complex general rectangular matrix f08bv nagf_lapackeig_ztzrzfReduces a complex upper trapezoidal matrix to upper triangular form f08bx nagf_lapackeig_zunmrzApplies the unitary transformation determined by f08bv f08ce nagf_lapackeig_dgeqlfQL factorization of real general rectangular matrix f08cf nagf_lapackeig_dorgqlForm all or part of orthogonal Q from QL factorization determined by f08ce f08cg nagf_lapackeig_dormqlApplies the orthogonal transformation determined by f08ce f08ch nagf_lapackeig_dgerqfRQ factorization of real general rectangular matrix f08cj nagf_lapackeig_dorgrqForm all or part of orthogonal Q from RQ factorization determined by f08ch f08ck nagf_lapackeig_dormrqApplies the orthogonal transformation determined by f08ch f08cs nagf_lapackeig_zgeqlfQL factorization of complex general rectangular matrix f08ct nagf_lapackeig_zungqlForm all or part of unitary Q from QL factorization determined by f08cs f08cu nagf_lapackeig_zunmqlApplies the unitary transformation determined by f08cs f08cv nagf_lapackeig_zgerqfRQ factorization of complex general rectangular matrix f08cw nagf_lapackeig_zungrqForm all or part of unitary Q from RQ factorization determined by f08cv f08cx nagf_lapackeig_zunmrqApplies the unitary transformation determined by f08cv f08fa nagf_lapackeig_dsyevComputes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fb nagf_lapackeig_dsyevxComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix f08fc nagf_lapackeig_dsyevdComputes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix (divide-and-conquer) f08fd nagf_lapackeig_dsyevrComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) f08fe nagf_lapackeig_dsytrdOrthogonal reduction of real symmetric matrix to symmetric tridiagonal form f08ff nagf_lapackeig_dorgtrGenerate orthogonal transformation matrix from reduction to tridiagonal form determined by f08fe f08fg nagf_lapackeig_dormtrApplies the orthogonal transformation determined by f08fe f08fl nagf_lapackeig_ddisnaComputes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix or for the left or right singular vectors of a general matrix f08fn nagf_lapackeig_zheevComputes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fp nagf_lapackeig_zheevxComputes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix f08fq nagf_lapackeig_zheevdComputes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix (divide-and-conquer) f08fr nagf_lapackeig_zheevrComputes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) f08fs nagf_lapackeig_zhetrdUnitary reduction of complex Hermitian matrix to real symmetric tridiagonal form f08ft nagf_lapackeig_zungtrGenerate unitary transformation matrix from reduction to tridiagonal form determined by f08fs f08fu nagf_lapackeig_zunmtrApplies the unitary transformation matrix determined by f08fs f08ga nagf_lapackeig_dspevComputes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gb nagf_lapackeig_dspevxComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage f08gc nagf_lapackeig_dspevdComputes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the QL or QR algorithm) f08ge nagf_lapackeig_dsptrdOrthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage f08gf nagf_lapackeig_dopgtrGenerate orthogonal transformation matrix from reduction to tridiagonal form determined by f08ge f08gg nagf_lapackeig_dopmtrApplies the orthogonal transformation determined by f08ge f08gn nagf_lapackeig_zhpevComputes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gp nagf_lapackeig_zhpevxComputes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage f08gq nagf_lapackeig_zhpevdComputes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the QL or QR algorithm) f08gs nagf_lapackeig_zhptrdPerforms a unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form, packed storage f08gt nagf_lapackeig_zupgtrGenerates a unitary transformation matrix from reduction to tridiagonal form determined by f08gs f08gu nagf_lapackeig_zupmtrApplies the unitary transformation matrix determined by f08gs f08ha nagf_lapackeig_dsbevComputes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hb nagf_lapackeig_dsbevxComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix f08hc nagf_lapackeig_dsbevdComputes all eigenvalues and, optionally, all eigenvectors of real symmetric band matrix (divide-and-conquer or Pal–Walker–Kahan variant of the QL or QR algorithm) f08he nagf_lapackeig_dsbtrdPerforms an orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form f08hn nagf_lapackeig_zhbevComputes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hp nagf_lapackeig_zhbevxComputes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix f08hq nagf_lapackeig_zhbevdComputes all eigenvalues and, optionally, all eigenvectors of complex Hermitian band matrix (divide-and-conquer) f08hs nagf_lapackeig_zhbtrdPerforms a unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form f08ja nagf_lapackeig_dstevComputes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jb nagf_lapackeig_dstevxComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix f08jc nagf_lapackeig_dstevdComputes all eigenvalues and, optionally, all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer) f08jd nagf_lapackeig_dstevrComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) f08je nagf_lapackeig_dsteqrComputes all eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using the implicit QL or QR algorithm f08jf nagf_lapackeig_dsterfComputes all eigenvalues of real symmetric tridiagonal matrix, root-free variant of the QL or QR algorithm f08jg nagf_lapackeig_dpteqrComputes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from real symmetric positive definite matrix f08jh nagf_lapackeig_dstedcComputes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) f08jj nagf_lapackeig_dstebzComputes selected eigenvalues of real symmetric tridiagonal matrix by bisection f08jk nagf_lapackeig_dsteinComputes selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array f08jl nagf_lapackeig_dstegrComputes selected eigenvalues and, optionally, the corresponding eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) f08js nagf_lapackeig_zsteqrComputes all eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using the implicit QL or QR algorithm f08ju nagf_lapackeig_zpteqrComputes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from complex Hermitian positive definite matrix f08jv nagf_lapackeig_zstedcComputes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) f08jx nagf_lapackeig_zsteinComputes selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array f08jy nagf_lapackeig_zstegrComputes selected eigenvalues and, optionally, the corresponding eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) f08ka nagf_lapackeig_dgelssComputes the minimum-norm solution to a real linear least squares problem using singular value decomposition f08kb nagf_lapackeig_dgesvdComputes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors f08kc nagf_lapackeig_dgelsdComputes the minimum-norm solution to a real linear least squares problem using singular value decomposition (divide-and-conquer) f08kd nagf_lapackeig_dgesddComputes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) f08ke nagf_lapackeig_dgebrdPerforms an orthogonal reduction of real general rectangular matrix to bidiagonal form f08kf nagf_lapackeig_dorgbrGenerates an orthogonal transformation matrices from reduction to bidiagonal form determined by f08ke f08kg nagf_lapackeig_dormbrApplies the orthogonal transformations from reduction to bidiagonal form determined by f08ke f08kh nagf_lapackeig_dgejsvComputes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (preconditioned Jacobi) f08kj nagf_lapackeig_dgesvjComputes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (fast Jacobi) f08kn nagf_lapackeig_zgelssComputes the minimum-norm solution to a complex linear least squares problem using singular value decomposition f08kp nagf_lapackeig_zgesvdComputes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors f08kq nagf_lapackeig_zgelsdComputes the minimum-norm solution to a complex linear least squares problem using singular value decomposition (divide-and-conquer) f08kr nagf_lapackeig_zgesddComputes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) f08ks nagf_lapackeig_zgebrdPerforms a unitary reduction of complex general rectangular matrix to bidiagonal form f08kt nagf_lapackeig_zungbrGenerates unitary transformation matrices from the reduction to bidiagonal form determined by f08ks f08ku nagf_lapackeig_zunmbrApplies the unitary transformations from reduction to bidiagonal form determined by f08ks f08le nagf_lapackeig_dgbbrdPerforms a reduction of real rectangular band matrix to upper bidiagonal form f08ls nagf_lapackeig_zgbbrdReduction of complex rectangular band matrix to upper bidiagonal form f08md nagf_lapackeig_dbdsdcComputes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) f08me nagf_lapackeig_dbdsqrPerforms an SVD of real bidiagonal matrix reduced from real general matrix f08ms nagf_lapackeig_zbdsqrPerforms an SVD of real bidiagonal matrix reduced from complex general matrix f08na nagf_lapackeig_dgeevComputes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix f08nb nagf_lapackeig_dgeevxComputes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors f08ne nagf_lapackeig_dgehrdPerforms an orthogonal reduction of real general matrix to upper Hessenberg form f08nf nagf_lapackeig_dorghrGenerates an orthogonal transformation matrix from reduction to Hessenberg form determined by f08ne f08ng nagf_lapackeig_dormhrApplies the orthogonal transformation matrix from reduction to Hessenberg form determined by f08ne f08nh nagf_lapackeig_dgebalBalances a real general matrix f08nj nagf_lapackeig_dgebakTransforms eigenvectors of real balanced matrix to those of original matrix supplied to f08nh f08nn nagf_lapackeig_zgeevComputes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix f08np nagf_lapackeig_zgeevxComputes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors f08ns nagf_lapackeig_zgehrdPerforms a unitary reduction of complex general matrix to upper Hessenberg form f08nt nagf_lapackeig_zunghrGenerates a unitary transformation matrix from reduction to Hessenberg form determined by f08ns f08nu nagf_lapackeig_zunmhrApplies the unitary transformation matrix from reduction to Hessenberg form determined by f08ns f08nv nagf_lapackeig_zgebalBalances a complex general matrix f08nw nagf_lapackeig_zgebakTransforms eigenvectors of complex balanced matrix to those of original matrix supplied to f08nv f08pa nagf_lapackeig_dgeesComputes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors f08pb nagf_lapackeig_dgeesxComputes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues f08pe nagf_lapackeig_dhseqrComputes the eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix f08pk nagf_lapackeig_dhseinComputes selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration f08pn nagf_lapackeig_zgeesComputes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors f08pp nagf_lapackeig_zgeesxComputes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also computes a reciprocal condition number for the average of the selected eigenvalues and for the right invariant subspace corresponding to these eigenvalues f08ps nagf_lapackeig_zhseqrComputes the eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix f08px nagf_lapackeig_zhseinComputes selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration f08qf nagf_lapackeig_dtrexcReorders a Schur factorization of real matrix using orthogonal similarity transformation f08qg nagf_lapackeig_dtrsenReorders a Schur factorization of real matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities f08qh nagf_lapackeig_dtrsylSolves the real Sylvester matrix equation AX+XB=C, A and B are upper quasi-triangular or transposes f08qk nagf_lapackeig_dtrevcComputes left and right eigenvectors of real upper quasi-triangular matrix f08ql nagf_lapackeig_dtrsnaComputes estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix f08qt nagf_lapackeig_ztrexcReorders a Schur factorization of complex matrix using unitary similarity transformation f08qu nagf_lapackeig_ztrsenReorders a Schur factorization of complex matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities f08qv nagf_lapackeig_ztrsylSolves the complex Sylvester matrix equation AX+XB=C, A and B are upper triangular or conjugate-transposes f08qx nagf_lapackeig_ztrevcComputes left and right eigenvectors of complex upper triangular matrix f08qy nagf_lapackeig_ztrsnaComputes estimates of sensitivities of selected eigenvalues and eigenvectors of complex upper triangular matrix f08ra nagf_lapackeig_dorcsdComputes the CS decomposition of an orthogonal matrix partitioned into four real submatrices f08rn nagf_lapackeig_zuncsdComputes the CS decomposition of a unitary matrix partitioned into four complex submatrices f08sa nagf_lapackeig_dsygvComputes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem f08sb nagf_lapackeig_dsygvxComputes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem f08sc nagf_lapackeig_dsygvdComputes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) f08se nagf_lapackeig_dsygstPerforms a reduction to standard form of real symmetric-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, B factorized by f07fd f08sn nagf_lapackeig_zhegvComputes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem f08sp nagf_lapackeig_zhegvxComputes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem f08sq nagf_lapackeig_zhegvdComputes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) f08ss nagf_lapackeig_zhegstPerforms a reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, B factorized by f07fr f08ta nagf_lapackeig_dspgvComputes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage f08tb nagf_lapackeig_dspgvxComputes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage f08tc nagf_lapackeig_dspgvdComputes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) f08te nagf_lapackeig_dspgstPerforms a reduction to standard form of real symmetric-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, packed storage, B factorized by f07gd f08tn nagf_lapackeig_zhpgvComputes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage f08tp nagf_lapackeig_zhpgvxComputes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage f08tq nagf_lapackeig_zhpgvdComputes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer) f08ts nagf_lapackeig_zhpgstPerforms a reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, packed storage, B factorized by f07gr f08ua nagf_lapackeig_dsbgvComputes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem f08ub nagf_lapackeig_dsbgvxComputes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem f08uc nagf_lapackeig_dsbgvdComputes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) f08ue nagf_lapackeig_dsbgstPerforms a reduction of real symmetric-definite banded generalized eigenproblem Ax=λBx to standard form Cy=λy, such that C has the same bandwidth as A f08uf nagf_lapackeig_dpbstfComputes a split Cholesky factorization of real symmetric positive definite band matrix A f08un nagf_lapackeig_zhbgvComputes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem f08up nagf_lapackeig_zhbgvxComputes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem f08uq nagf_lapackeig_zhbgvdComputes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) f08us nagf_lapackeig_zhbgstPerforms a reduction of complex Hermitian-definite banded generalized eigenproblem Ax=λBx to standard form Cy=λy, such that C has the same bandwidth as A f08ut nagf_lapackeig_zpbstfComputes a split Cholesky factorization of complex Hermitian positive definite band matrix A f08va nagf_lapackeig_dggsvdComputes the generalized singular value decomposition of a real matrix pair f08vc nagf_lapackeig_dggsvd3Computes, using BLAS-3, the generalized singular value decomposition of a real matrix pair f08ve nagf_lapackeig_dggsvpProduces orthogonal matrices that simultaneously reduce the m by n matrix A and the p by n matrix B to upper triangular form f08vg nagf_lapackeig_dggsvp3Produces orthogonal matrices, using BLAS-3, that simultaneously reduce the m by n matrix A and the p by n matrix B to upper triangular form f08vn nagf_lapackeig_zggsvdComputes the generalized singular value decomposition of a complex matrix pair f08vq nagf_lapackeig_zggsvd3Computes, using BLAS-3, the generalized singular value decomposition of a complex matrix pair f08vs nagf_lapackeig_zggsvpProduces unitary matrices that simultaneously reduce the complex, m by n, matrix A and the complex, p by n, matrix B to upper triangular form f08vu nagf_lapackeig_zggsvp3Produces unitary matrices, using BLAS-3, that simultaneously reduce the complex, m by n, matrix A and the complex, p by n, matrix B to upper triangular form f08wa nagf_lapackeig_dggevComputes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wb nagf_lapackeig_dggevxComputes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors f08wc nagf_lapackeig_dggev3Computes, for a real nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08we nagf_lapackeig_dgghrdPerforms an orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form f08wf nagf_lapackeig_dgghd3Performs, using BLAS-3, an orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form f08wh nagf_lapackeig_dggbalBalances a pair of real, square, matrices f08wj nagf_lapackeig_dggbakTransforms eigenvectors of a pair of real balanced matrices to those of original matrix pair supplied to f08wh f08wn nagf_lapackeig_zggevComputes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08wp nagf_lapackeig_zggevxComputes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors f08wq nagf_lapackeig_zggev3Computes, for a complex nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors f08ws nagf_lapackeig_zgghrdPerforms a unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form f08wt nagf_lapackeig_zgghd3Performs, using BLAS-3, a unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form f08wv nagf_lapackeig_zggbalBalances a pair of complex, square, matrices f08ww nagf_lapackeig_zggbakTransforms eigenvectors of a pair of complex balanced matrices to those of original matrix pair supplied to f08wv f08xa nagf_lapackeig_dggesComputes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors f08xb nagf_lapackeig_dggesxComputes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues f08xc nagf_lapackeig_dgges3Computes, for a real nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors f08xe nagf_lapackeig_dhgeqzComputes eigenvalues and generalized Schur factorization of real generalized upper Hessenberg form reduced from a pair of real general matrices f08xn nagf_lapackeig_zggesComputes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors f08xp nagf_lapackeig_zggesxComputes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues f08xq nagf_lapackeig_zgges3Computes, for a complex nonsymmetric matrix pair, using BLAS-3, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors f08xs nagf_lapackeig_zhgeqzEigenvalues and generalized Schur factorization of complex generalized upper Hessenberg form reduced from a pair of complex, square, matrices f08ye nagf_lapackeig_dtgsjaComputes the generalized singular value decomposition of a real upper triangular (or trapezoidal) matrix pair f08yf nagf_lapackeig_dtgexcReorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation f08yg nagf_lapackeig_dtgsenReorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces f08yh nagf_lapackeig_dtgsylSolves the real-valued, generalized, quasi-trangular, Sylvester equation f08yk nagf_lapackeig_dtgevcComputes right and left generalized eigenvectors of the matrix pair (A,B) which is assumed to be in generalized upper Schur form f08yl nagf_lapackeig_dtgsnaEstimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a real matrix pair in generalized real Schur canonical form f08ys nagf_lapackeig_ztgsjaComputes the generalized singular value decomposition of a complex upper triangular (or trapezoidal) matrix pair f08yt nagf_lapackeig_ztgexcReorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation f08yu nagf_lapackeig_ztgsenReorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces f08yv nagf_lapackeig_ztgsylSolves the complex generalized Sylvester equation f08yx nagf_lapackeig_ztgevcComputes left and right eigenvectors of a pair of complex upper triangular matrices f08yy nagf_lapackeig_ztgsnaEstimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur canonical form f08za nagf_lapackeig_dgglseSolves the real linear equality-constrained least squares (LSE) problem f08zb nagf_lapackeig_dggglmSolves a real general Gauss–Markov linear model (GLM) problem f08ze nagf_lapackeig_dggqrfComputes a generalized QR factorization of a real matrix pair f08zf nagf_lapackeig_dggrqfComputes a generalized RQ factorization of a real matrix pair f08zn nagf_lapackeig_zgglseSolves the complex linear equality-constrained least squares (LSE) problem f08zp nagf_lapackeig_zggglmSolves a complex general Gauss–Markov linear model (GLM) problem f08zs nagf_lapackeig_zggqrfComputes a generalized QR factorization of a complex matrix pair f08zt nagf_lapackeig_zggrqfComputes a generalized RQ factorization of a complex matrix pair
Back to F08 Chapter Summary

### F10 – Randomized Numerical Linear Algebra

Examples of routines and methods in this chapter:

Back to F10 Chapter Summary

### F11 – Large Scale Linear Systems

Examples of routines and methods in this chapter:

 f11bd nagf_sparse_real_gen_basic_setupReal sparse nonsymmetric linear systems, setup for f11be f11be nagf_sparse_real_gen_basic_solverReal sparse nonsymmetric linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method f11bf nagf_sparse_real_gen_basic_diagReal sparse nonsymmetric linear systems, diagnostic for f11be f11br nagf_sparse_complex_gen_basic_setupComplex sparse non-Hermitian linear systems, setup for f11bs f11bs nagf_sparse_complex_gen_basic_solverComplex sparse non-Hermitian linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method f11bt nagf_sparse_complex_gen_basic_diagComplex sparse non-Hermitian linear systems, diagnostic for f11bs f11da nagf_sparse_real_gen_precon_iluReal sparse nonsymmetric linear systems, incomplete LU factorization f11db nagf_sparse_real_gen_precon_ilu_solveSolution of linear system involving incomplete LU preconditioning matrix generated by f11da f11dc nagf_sparse_real_gen_solve_iluSolution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by f11da f11dd nagf_sparse_real_gen_precon_ssor_solveSolution of linear system involving preconditioning matrix generated by applying SSOR to real sparse nonsymmetric matrix f11de nagf_sparse_real_gen_solve_jacssorSolution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB, or TFQMR method, Jacobi or SSOR preconditioner (Black Box) f11df nagf_sparse_real_gen_precon_bdiluReal sparse nonsymmetric linear system, incomplete LU factorization of local or overlapping diagonal blocks f11dg nagf_sparse_real_gen_solve_bdiluSolution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete LU block diagonal preconditioner computed by f11df f11dk nagf_sparse_real_gen_precon_jacobiReal, sparse, symmetric or nonsymmetric, linear systems, line Jacobi preconditioner f11dn nagf_sparse_complex_gen_precon_iluComplex sparse non-Hermitian linear systems, incomplete LU factorization f11dp nagf_sparse_complex_gen_precon_ilu_solveSolution of complex linear system involving incomplete LU preconditioning matrix generated by f11dn f11dq nagf_sparse_complex_gen_solve_iluSolution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by f11dn (Black Box) f11dr nagf_sparse_complex_gen_precon_ssor_solveSolution of linear system involving preconditioning matrix generated by applying SSOR to complex sparse non-Hermitian matrix f11ds nagf_sparse_complex_gen_solve_jacssorSolution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, Jacobi or SSOR preconditioner Black Box f11dt nagf_sparse_complex_gen_precon_bdiluComplex, sparse, non-Hermitian linear system, incomplete LU factorization of local or overlapping diagonal blocks f11du nagf_sparse_complex_gen_solve_bdiluSolution of complex, sparse, non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete LU block diagonal preconditioner computed by f11dt f11dx nagf_sparse_complex_gen_precon_jacobiComplex, sparse, Hermitian or non-Hermitian, linear systems, line Jacobi preconditioner f11gd nagf_sparse_real_symm_basic_setupReal sparse symmetric linear systems, setup for f11ge f11ge nagf_sparse_real_symm_basic_solverReal sparse symmetric linear systems, preconditioned conjugate gradient or Lanczos method or the MINRES algorithm f11gf nagf_sparse_real_symm_basic_diagReal sparse symmetric linear systems, diagnostic for f11ge f11gr nagf_sparse_complex_herm_basic_setupComplex sparse Hermitian linear systems, setup for f11gs f11gs nagf_sparse_complex_herm_basic_solverComplex sparse Hermitian linear systems, preconditioned conjugate gradient or Lanczos f11gt nagf_sparse_complex_herm_basic_diagComplex sparse Hermitian linear systems, diagnostic for f11gs f11ja nagf_sparse_real_symm_precon_icholReal sparse symmetric matrix, incomplete Cholesky factorization f11jb nagf_sparse_real_symm_precon_ichol_solveSolution of linear system involving incomplete Cholesky preconditioning matrix generated by f11ja f11jc nagf_sparse_real_symm_solve_icholSolution of real sparse symmetric linear system, conjugate gradient/Lanczos method, preconditioner computed by f11ja (Black Box) f11jd nagf_sparse_real_symm_precon_ssor_solveSolution of linear system involving preconditioning matrix generated by applying SSOR to real sparse symmetric matrix f11je nagf_sparse_real_symm_solve_jacssorSolution of real sparse symmetric linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) f11jn nagf_sparse_complex_herm_precon_icholComplex sparse Hermitian matrix, incomplete Cholesky factorization f11jp nagf_sparse_complex_herm_precon_ilu_solveSolution of complex linear system involving incomplete Cholesky preconditioning matrix generated by f11jn f11jq nagf_sparse_complex_herm_solve_iluSolution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, preconditioner computed by f11jn (Black Box) f11jr nagf_sparse_complex_herm_precon_ssor_solveSolution of linear system involving preconditioning matrix generated by applying SSOR to complex sparse Hermitian matrix f11js nagf_sparse_complex_herm_solve_jacssorSolution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) f11md nagf_sparse_direct_real_gen_setupReal sparse nonsymmetric linear systems, setup for f11me f11me nagf_sparse_direct_real_gen_luLU factorization of real sparse matrix f11mf nagf_sparse_direct_real_gen_solveSolution of real sparse simultaneous linear equations (coefficient matrix already factorized) f11mg nagf_sparse_direct_real_gen_condEstimate condition number of real matrix, matrix already factorized by f11me f11mh nagf_sparse_direct_real_gen_refineRefined solution with error bounds of real system of linear equations, multiple right-hand sides f11mk nagf_sparse_direct_real_gen_matmulReal sparse nonsymmetric matrix-matrix multiply, compressed column storage f11ml nagf_sparse_direct_real_gen_norm1-norm, ∞-norm, largest absolute element, real, square, sparse matrix f11mm nagf_sparse_direct_real_gen_diagReal sparse nonsymmetric linear systems, diagnostic for f11me f11xa nagf_sparse_real_gen_matvecReal, sparse, nonsymmetric matrix-vector multiply f11xe nagf_sparse_real_symm_matvecReal sparse symmetric matrix-vector multiply f11xn nagf_sparse_complex_gen_matvecComplex sparse non-Hermitian matrix-vector multiply f11xs nagf_sparse_complex_herm_matvecComplex sparse Hermitian matrix-vector multiply f11ye nagf_sparse_sym_rcmReverse Cuthill–McKee reordering of a sparse symmetric matrix in CCS format f11za nagf_sparse_real_gen_sortReal sparse nonsymmetric matrix reorder routine f11zb nagf_sparse_real_symm_sortReal sparse symmetric matrix reorder routine f11zn nagf_sparse_complex_gen_sortComplex sparse non-Hermitian matrix reorder routine f11zp nagf_sparse_complex_herm_sortComplex sparse Hermitian matrix reorder routine
Back to F11 Chapter Summary

### F12 – Large Scale Eigenproblems

Examples of routines and methods in this chapter:

 f12aa nagf_sparseig_real_initInitialization routine for (f12ab) computing selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric sparse (standard or generalized) eigenproblem f12ab nagf_sparseig_real_iterSelected eigenvalues and, optionally, eigenvectors of a real nonsymmetric sparse eigenproblem, reverse communication f12ac nagf_sparseig_real_procSelected eigenvalues and, optionally, eigenvectors of a real nonsymmetric sparse eigenproblem, postprocessing for f12ab f12ad nagf_sparseig_real_optionSet a single option from a string (f12ab/f12ac/f12ag) f12ae nagf_sparseig_real_monitProvides monitoring information for f12ab f12af nagf_sparseig_real_band_initInitialization routine for (f12ag) computing selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric banded (standard or generalized) eigenproblem f12ag nagf_sparseig_real_band_solveSelected eigenvalues and, optionally, eigenvectors of a real nonsymmetric banded eigenproblem, driver f12an nagf_sparseig_complex_initInitialization routine for (f12ap) computing selected eigenvalues and, optionally, eigenvectors of a complex sparse (standard or generalized) eigenproblem f12ap nagf_sparseig_complex_iterSelected eigenvalues and, optionally, eigenvectors of a complex sparse eigenproblem, reverse communication f12aq nagf_sparseig_complex_procSelected eigenvalues and, optionally, eigenvectors of a complex sparse eigenproblem, postprocessing for f12ap f12ar nagf_sparseig_complex_optionSet a single option from a string (f12ap/f12aq) f12as nagf_sparseig_complex_monitProvides monitoring information for f12ap f12at nagf_sparseig_complex_band_initInitialization routine for f12au computing selected eigenvalues and, optionally, eigenvectors of a complex banded (standard or generalized) eigenproblem f12au nagf_sparseig_complex_band_solveSelected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver f12fa nagf_sparseig_real_symm_initInitialization routine for (f12fb) computing selected eigenvalues and, optionally, eigenvectors of a real symmetric sparse (standard or generalized) eigenproblem f12fb nagf_sparseig_real_symm_iterSelected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, reverse communication f12fc nagf_sparseig_real_symm_procSelected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, postprocessing for f12fb f12fd nagf_sparseig_real_symm_optionSet a single option from a string (f12fb/f12fc/f12fg) f12fe nagf_sparseig_real_symm_monitProvides monitoring information for f12fb f12ff nagf_sparseig_real_symm_band_initInitialization routine for (f12fg) computing selected eigenvalues and, optionally, eigenvectors of a real symmetric banded (standard or generalized) eigenproblem f12fg nagf_sparseig_real_symm_band_solveSelected eigenvalues and, optionally, eigenvectors of a real symmetric banded eigenproblem, driver
Back to F12 Chapter Summary

### F16 – Further Linear Algebra Support Routines

Examples of routines and methods in this chapter:

 f16dl nagf_blast_isumSum elements of integer vector f16dn nagf_blast_imax_valMaximum value and location, integer vector f16dp nagf_blast_imin_valMinimum value and location, integer vector f16dq nagf_blast_iamax_valMaximum absolute value and location, integer vector f16dr nagf_blast_iamin_valMinimum absolute value and location, integer vector f16ea nagf_blast_ddotDot product of two vectors, allows scaling and accumulation f16ec nagf_blast_daxpbyReal weighted vector addition f16eh nagf_blast_dwaxpbyReal weighted vector addition preserving input f16el nagf_blast_dsumSum elements of real vector f16gc nagf_blast_zaxpbyComplex weighted vector addition f16gh nagf_blast_zwaxpbyComplex weighted vector addition preserving input f16gl nagf_blast_zsumSum elements of complex vector f16jn nagf_blast_dmax_valMaximum value and location, real vector f16jp nagf_blast_dmin_valMinimum value and location, real vector f16jq nagf_blast_damax_valMaximum absolute value and location, real vector f16jr nagf_blast_damin_valMinimum absolute value and location, real vector f16js nagf_blast_zamax_valMaximum absolute value and location, complex vector f16jt nagf_blast_zamin_valMinimum absolute value and location, complex vector f16rb nagf_blast_dgb_norm1-norm, ∞-norm, Frobenius norm, largest absolute element, real band matrix f16ub nagf_blast_zgb_norm1-norm, ∞-norm, Frobenius norm, largest absolute element, complex band matrix
Back to F16 Chapter Summary

### G01 – Simple Calculations on Statistical Data

Examples of routines and methods in this chapter:

 g01ab nagf_stat_summary_2varMeans, corrected sums of squares and cross-products, etc., two variables, from raw data g01ad nagf_stat_summary_freqMean, variance, skewness, kurtosis, etc., one variable, from frequency table g01ae nagf_stat_frequency_tableFrequency table from raw data g01af nagf_stat_contingency_tableTwo-way contingency table analysis, with χ2/Fisher's exact test g01al nagf_stat_5pt_summaryComputes a five-point summary (median, hinges and extremes) g01am nagf_stat_quantilesFind quantiles of an unordered vector, real numbers g01an nagf_stat_quantiles_stream_fixedCalculates approximate quantiles from a data stream of known size g01ap nagf_stat_quantiles_stream_arbitraryCalculates approximate quantiles from a data stream of unknown size g01ar nagf_stat_plot_stem_leafConstructs a stem and leaf plot g01as nagf_stat_plot_box_whiskerConstructs a box and whisker plot g01at nagf_stat_summary_onevarComputes univariate summary information: mean, variance, skewness, kurtosis g01au nagf_stat_summary_onevar_combineCombines multiple sets of summary information, for use after g01at g01bj nagf_stat_prob_binomialBinomial distribution function g01bk nagf_stat_prob_poissonPoisson distribution function g01bl nagf_stat_prob_hypergeomHypergeometric distribution function g01da nagf_stat_normal_scores_exactNormal scores, accurate values g01db nagf_stat_normal_scores_approxNormal scores, approximate values g01dc nagf_stat_normal_scores_varNormal scores, approximate variance-covariance matrix g01dd nagf_stat_test_shapiro_wilkShapiro and Wilk's W test for Normality g01dh nagf_stat_ranks_and_scoresRanks, Normal scores, approximate Normal scores or exponential (Savage) scores g01ea nagf_stat_prob_normalComputes probabilities for the standard Normal distribution g01eb nagf_stat_prob_students_tComputes probabilities for Student's t-distribution g01ec nagf_stat_prob_chisqComputes probabilities for χ2 distribution g01ed nagf_stat_prob_fComputes probabilities for F-distribution g01ee nagf_stat_prob_betaComputes upper and lower tail probabilities and probability density function for the beta distribution g01ef nagf_stat_prob_gammaComputes probabilities for the gamma distribution g01em nagf_stat_prob_studentized_rangeComputes probability for the Studentized range statistic g01ep nagf_stat_prob_durbin_watsonComputes bounds for the significance of a Durbin–Watson statistic g01er nagf_stat_prob_vonmisesComputes probability for von Mises distribution g01et nagf_stat_prob_landauLandau distribution function g01eu nagf_stat_prob_vavilovVavilov distribution function g01ew nagf_stat_prob_dickey_fuller_unitComputes probabilities for the Dickey–Fuller unit root test g01ey nagf_stat_prob_kolmogorov1Computes probabilities for the one-sample Kolmogorov–Smirnov distribution g01ez nagf_stat_prob_kolmogorov2Computes probabilities for the two-sample Kolmogorov–Smirnov distribution g01fa nagf_stat_inv_cdf_normalComputes deviates for the standard Normal distribution g01fb nagf_stat_inv_cdf_students_tComputes deviates for Student's t-distribution g01fc nagf_stat_inv_cdf_chisqComputes deviates for the χ2 distribution g01fd nagf_stat_inv_cdf_fComputes deviates for the F-distribution g01fe nagf_stat_inv_cdf_betaComputes deviates for the beta distribution g01ff nagf_stat_inv_cdf_gammaComputes deviates for the gamma distribution g01fm nagf_stat_inv_cdf_studentized_rangeComputes deviates for the Studentized range statistic g01ft nagf_stat_inv_cdf_landauLandau inverse function Ψ(x) g01gb nagf_stat_prob_students_t_noncentralComputes probabilities for the non-central Student's t-distribution g01gc nagf_stat_prob_chisq_noncentralComputes probabilities for the non-central χ2 distribution g01gd nagf_stat_prob_f_noncentralComputes probabilities for the non-central F-distribution g01ge nagf_stat_prob_beta_noncentralComputes probabilities for the non-central beta distribution g01ha nagf_stat_prob_bivariate_normalComputes probability for the bivariate Normal distribution g01hb nagf_stat_prob_multi_normalComputes probabilities for the multivariate Normal distribution g01hc nagf_stat_prob_bivariate_students_tComputes probabilities for the bivariate Student's t-distribution g01hd nagf_stat_prob_multi_students_tComputes the probability for the multivariate Student's t-distribution g01jc nagf_stat_prob_chisq_noncentral_lincombComputes probability for a positive linear combination of χ2 variables g01jd nagf_stat_prob_chisq_lincombComputes lower tail probability for a linear combination of (central) χ2 variables g01ka nagf_stat_pdf_normalCalculates the value for the probability density function of the Normal distribution at a chosen point g01kf nagf_stat_pdf_gammaCalculates the value for the probability density function of the gamma distribution at a chosen point g01kk nagf_stat_pdf_gamma_vectorComputes a vector of values for the probability density function of the gamma distribution g01kq nagf_stat_pdf_normal_vectorComputes a vector of values for the probability density function of the Normal distribution g01lb nagf_stat_pdf_multi_normal_vectorComputes a vector of values for the probability density function of the multivariate Normal distribution g01mb nagf_stat_mills_ratioComputes reciprocal of Mills' Ratio g01mt nagf_stat_pdf_landauLandau density function ϕ(λ) g01mu nagf_stat_pdf_vavilovVavilov density function ϕV(λ;κ,β2) g01na nagf_stat_moments_quad_formCumulants and moments of quadratic forms in Normal variables g01nb nagf_stat_moments_ratio_quad_formsMoments of ratios of quadratic forms in Normal variables, and related statistics g01pt nagf_stat_pdf_landau_moment1Landau first moment function Φ1(x) g01qt nagf_stat_pdf_landau_moment2Landau second moment function Φ2(x) g01rt nagf_stat_pdf_landau_derivLandau derivative function ϕ′(λ) g01sa nagf_stat_prob_normal_vectorComputes a vector of probabilities for the standard Normal distribution g01sb nagf_stat_prob_students_t_vectorComputes a vector of probabilities for the Student's t-distribution g01sc nagf_stat_prob_chisq_vectorComputes a vector of probabilities for χ2 distribution g01sd nagf_stat_prob_f_vectorComputes a vector of probabilities for F-distribution g01se nagf_stat_prob_beta_vectorComputes a vector of probabilities for the beta distribution g01sf nagf_stat_prob_gamma_vectorComputes a vector of probabilities for the gamma distribution g01sj nagf_stat_prob_binomial_vectorComputes a vector of probabilities for the binomial distribution g01sk nagf_stat_prob_poisson_vectorComputes a vector of probabilities for the Poisson distribution g01sl nagf_stat_prob_hypergeom_vectorComputes a vector of probabilities for the hypergeometric distribution g01ta nagf_stat_inv_cdf_normal_vectorComputes a vector of deviates for the standard Normal distribution g01tb nagf_stat_inv_cdf_students_t_vectorComputes a vector of deviates for Student's t-distribution g01tc nagf_stat_inv_cdf_chisq_vectorComputes a vector of deviates for χ2 distribution g01td nagf_stat_inv_cdf_f_vectorComputes a vector of deviates for F-distribution g01te nagf_stat_inv_cdf_beta_vectorComputes a vector of deviates for the beta distribution g01tf nagf_stat_inv_cdf_gamma_vectorComputes a vector of deviates for the gamma distribution g01wa nagf_stat_moving_averageComputes the mean and standard deviation using a rolling window g01zu nagf_stat_init_vavilovInitialization routine for g01mu and g01eu
Back to G01 Chapter Summary

### G02 – Correlation and Regression Analysis

Examples of routines and methods in this chapter:

Back to G02 Chapter Summary

### G03 – Multivariate Methods

Examples of routines and methods in this chapter:

 g03aa nagf_mv_prin_compPerforms principal component analysis g03ac nagf_mv_canon_varPerforms canonical variate analysis g03ad nagf_mv_canon_corrPerforms canonical correlation analysis g03ba nagf_mv_rot_orthomaxComputes orthogonal rotations for loading matrix, generalized orthomax criterion g03bc nagf_mv_rot_procrustesComputes Procrustes rotations g03bd nagf_mv_rot_promaxProMax rotations g03ca nagf_mv_factorComputes maximum likelihood estimates of the parameters of a factor analysis model, factor loadings, communalities and residual correlations g03cc nagf_mv_factor_scoreComputes factor score coefficients (for use after g03ca) g03da nagf_mv_discrimComputes test statistic for equality of within-group covariance matrices and matrices for discriminant analysis g03db nagf_mv_discrim_mahalComputes Mahalanobis squared distances for group or pooled variance-covariance matrices (for use after g03da) g03dc nagf_mv_discrim_groupAllocates observations to groups according to selected rules (for use after g03da) g03ea nagf_mv_distance_matComputes distance matrix g03ec nagf_mv_cluster_hierHierarchical cluster analysis g03ef nagf_mv_cluster_kmeansK-means cluster analysis g03eh nagf_mv_cluster_hier_dendrogramConstructs dendrogram (for use after g03ec) g03ej nagf_mv_cluster_hier_indicatorComputes cluster indicator variable (for use after g03ec) g03fa nagf_mv_multidimscal_metricPerforms principal coordinate analysis, classical metric scaling g03fc nagf_mv_multidimscal_ordinalPerforms non-metric (ordinal) multidimensional scaling g03ga nagf_mv_gaussian_mixtureFits a Gaussian mixture model g03za nagf_mv_z_scoresProduces standardized values (z-scores) for a data matrix
Back to G03 Chapter Summary

### G04 – Analysis of Variance

Examples of routines and methods in this chapter:

 g04ag nagf_anova_hier2Two-way analysis of variance, hierarchical classification, subgroups of unequal size g04bb nagf_anova_randomAnalysis of variance, randomized block or completely randomized design, treatment means and standard errors g04bc nagf_anova_rowcolAnalysis of variance, general row and column design, treatment means and standard errors g04ca nagf_anova_factorialAnalysis of variance, complete factorial design, treatment means and standard errors g04da nagf_anova_contrastsComputes sum of squares for contrast between means g04db nagf_anova_confidenceComputes confidence intervals for differences between means computed by g04bb or g04bc g04ea nagf_anova_dummyvarsComputes orthogonal polynomials or dummy variables for factor/classification variable g04ga nagf_anova_iccIntraclass correlation (ICC) for assessing rater reliability
Back to G04 Chapter Summary

### G05 – Random Number Generators

Examples of routines and methods in this chapter:

 g05kf nagf_rand_init_repeatInitializes a pseudorandom number generator to give a repeatable sequence g05kg nagf_rand_init_nonrepeatInitializes a pseudorandom number generator to give a non-repeatable sequence g05kh nagf_rand_init_leapfrogPrimes a pseudorandom number generator for generating multiple streams using leap-frog g05kj nagf_rand_init_skipaheadPrimes a pseudorandom number generator for generating multiple streams using skip-ahead g05kk nagf_rand_init_skipahead_power2Primes a pseudorandom number generator for generating multiple streams using skip-ahead, skipping ahead a power of 2 g05nc nagf_rand_permutePseudorandom permutation of an integer vector g05nd nagf_rand_samplePseudorandom sample from an integer vector g05ne nagf_rand_sample_wgtPseudorandom sample, without replacement, unequal weights g05pd nagf_rand_times_garch_asym1Generates a realization of a time series from a GARCH process with asymmetry of the form (εt-1+γ)2 g05pe nagf_rand_times_garch_asym2Generates a realization of a time series from a GARCH process with asymmetry of the form (|εt-1|+γ⁢εt-1)2 g05pf nagf_rand_times_garch_gjrGenerates a realization of a time series from an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process g05pg nagf_rand_times_garch_expGenerates a realization of a time series from an exponential GARCH (EGARCH) process g05ph nagf_rand_times_armaGenerates a realization of a time series from an ARMA model g05pj nagf_rand_times_mv_varmaGenerates a realization of a multivariate time series from a VARMA model g05pm nagf_rand_times_smooth_expGenerates a realization of a time series from an exponential smoothing model g05pv nagf_rand_kfold_xywPermutes a matrix, vector, vector triplet into a form suitable for K-fold cross validation g05pw nagf_rand_subsamp_xywPermutes a matrix, vector, vector triplet into a form suitable for random sub-sampling validation g05px nagf_rand_matrix_orthogGenerates a random orthogonal matrix g05py nagf_rand_matrix_corrGenerates a random correlation matrix g05pz nagf_rand_matrix_2waytableGenerates a random two-way table g05rc nagf_rand_copula_students_tGenerates a matrix of pseudorandom numbers from a Student's t-copula g05rd nagf_rand_copula_normalGenerates a matrix of pseudorandom numbers from a Gaussian copula g05re nagf_rand_copula_clayton_bivarGenerates a matrix of pseudorandom numbers from a bivariate Clayton/Cook–Johnson copula g05rf nagf_rand_copula_frank_bivarGenerates a matrix of pseudorandom numbers from a bivariate Frank copula g05rg nagf_rand_copula_plackett_bivarGenerates a matrix of pseudorandom numbers from a bivariate Plackett copula g05rh nagf_rand_copula_claytonGenerates a matrix of pseudorandom numbers from a multivariate Clayton/Cook–Johnson copula g05rj nagf_rand_copula_frankGenerates a matrix of pseudorandom numbers from a multivariate Frank copula g05rk nagf_rand_copula_gumbelGenerates a matrix of pseudorandom numbers from a Gumbel–Hougaard copula g05ry nagf_rand_multivar_students_tGenerates a matrix of pseudorandom numbers from a multivariate Student's t-distribution g05rz nagf_rand_multivar_normalGenerates a matrix of pseudorandom numbers from a multivariate Normal distribution g05sa nagf_rand_dist_uniform01Generates a vector of pseudorandom numbers from a uniform distribution over (0,1] g05sb nagf_rand_dist_betaGenerates a vector of pseudorandom numbers from a beta distribution g05sc nagf_rand_dist_cauchyGenerates a vector of pseudorandom numbers from a Cauchy distribution g05sd nagf_rand_dist_chisqGenerates a vector of pseudorandom numbers from a χ2 distribution g05se nagf_rand_dist_dirichletGenerates a vector of pseudorandom numbers from a Dirichlet distribution g05sf nagf_rand_dist_expGenerates a vector of pseudorandom numbers from an exponential distribution g05sg nagf_rand_dist_expmixGenerates a vector of pseudorandom numbers from an exponential mix distribution g05sh nagf_rand_dist_fGenerates a vector of pseudorandom numbers from an F-distribution g05sj nagf_rand_dist_gammaGenerates a vector of pseudorandom numbers from a gamma distribution g05sk nagf_rand_dist_normalGenerates a vector of pseudorandom numbers from a Normal distribution g05sl nagf_rand_dist_logisticGenerates a vector of pseudorandom numbers from a logistic distribution g05sm nagf_rand_dist_lognormalGenerates a vector of pseudorandom numbers from a log-normal distribution g05sn nagf_rand_dist_students_tGenerates a vector of pseudorandom numbers from a Student's t-distribution g05sp nagf_rand_dist_triangularGenerates a vector of pseudorandom numbers from a triangular distribution g05sq nagf_rand_dist_uniformGenerates a vector of pseudorandom numbers from a uniform distribution over [a,b] g05sr nagf_rand_dist_vonmisesGenerates a vector of pseudorandom numbers from a von Mises distribution g05ss nagf_rand_dist_weibullGenerates a vector of pseudorandom numbers from a Weibull distribution g05ta nagf_rand_int_binomialGenerates a vector of pseudorandom integers from a binomial distribution g05tb nagf_rand_logicalGenerates a vector of pseudorandom logical values g05tc nagf_rand_int_geomGenerates a vector of pseudorandom integers from a geometric distribution g05td nagf_rand_int_generalGenerates a vector of pseudorandom integers from a general discrete distribution g05te nagf_rand_int_hypergeomGenerates a vector of pseudorandom integers from a hypergeometric distribution g05tf nagf_rand_int_logGenerates a vector of pseudorandom integers from a logarithmic distribution g05tg nagf_rand_int_multinomialGenerates a vector of pseudorandom integers from a multinomial distribution g05th nagf_rand_int_negbinGenerates a vector of pseudorandom integers from a negative binomial distribution g05tj nagf_rand_int_poissonGenerates a vector of pseudorandom integers from a Poisson distribution g05tk nagf_rand_int_poisson_varmeanGenerates a vector of pseudorandom integers from a Poisson distribution with varying mean g05tl nagf_rand_int_uniformGenerates a vector of pseudorandom integers from a uniform distribution g05xa nagf_rand_bb_initInitializes the Brownian bridge generator g05xb nagf_rand_bbGenerate paths for a free or non-free Wiener process using the Brownian bridge algorithm g05xc nagf_rand_bb_inc_initInitializes the generator which backs out the increments of sample paths generated by a Brownian bridge algorithm g05xd nagf_rand_bb_incBacks out the increments from sample paths generated by a Brownian bridge algorithm g05xe nagf_rand_bb_make_bridge_orderCreates a Brownian bridge construction order out of a set of input times g05yj nagf_rand_quasi_normalGenerates a Normal quasi-random number sequence g05yk nagf_rand_quasi_lognormalGenerates a log-normal quasi-random number sequence g05yl nagf_rand_quasi_initInitializes a quasi-random number generator g05ym nagf_rand_quasi_uniformGenerates a uniform quasi-random number sequence g05yn nagf_rand_quasi_init_scrambledInitializes a scrambled quasi-random number generator g05zm nagf_rand_field_1d_user_setupSetup for simulating one-dimensional random fields, user-defined variogram g05zn nagf_rand_field_1d_predef_setupSetup for simulating one-dimensional random fields g05zp nagf_rand_field_1d_generateGenerates realizations of a one-dimensional random field g05zq nagf_rand_field_2d_user_setupSetup for simulating two-dimensional random fields, user-defined variogram g05zr nagf_rand_field_2d_predef_setupSetup for simulating two-dimensional random fields, preset variogram g05zs nagf_rand_field_2d_generateGenerates realizations of a two-dimensional random field g05zt nagf_rand_field_fracbm_generateGenerates realizations of fractional Brownian motion
Back to G05 Chapter Summary

### G07 – Univariate Estimation

Examples of routines and methods in this chapter:

 g07aa nagf_univar_ci_binomialComputes confidence interval for the parameter of a binomial distribution g07ab nagf_univar_ci_poissonComputes confidence interval for the parameter of a Poisson distribution g07bb nagf_univar_estim_normalComputes maximum likelihood estimates for parameters of the Normal distribution from grouped and/or censored data g07be nagf_univar_estim_weibullComputes maximum likelihood estimates for parameters of the Weibull distribution g07bf nagf_univar_estim_genparetoEstimates parameter values of the generalized Pareto distribution g07ca nagf_univar_ttest_2normalComputes t-test statistic for a difference in means between two Normal populations, confidence interval g07da nagf_univar_robust_1var_medianRobust estimation, median, median absolute deviation, robust standard deviation g07db nagf_univar_robust_1var_mestimRobust estimation, M-estimates for location and scale parameters, standard weight functions g07dc nagf_univar_robust_1var_mestim_wgtRobust estimation, M-estimates for location and scale parameters, user-defined weight functions g07dd nagf_univar_robust_1var_trimmedComputes a trimmed and winsorized mean of a single sample with estimates of their variance g07ea nagf_univar_robust_1var_ciRobust confidence intervals, one-sample g07eb nagf_univar_robust_2var_ciRobust confidence intervals, two-sample g07ga nagf_univar_outlier_peirce_1varOutlier detection using method of Peirce, raw data or single variance supplied g07gb nagf_univar_outlier_peirce_2varOutlier detection using method of Peirce, two variances supplied
Back to G07 Chapter Summary

### G08 – Nonparametric Statistics

Examples of routines and methods in this chapter:

 g08aa nagf_nonpar_test_signSign test on two paired samples g08ac nagf_nonpar_test_medianMedian test on two samples of unequal size g08ae nagf_nonpar_test_friedmanFriedman two-way analysis of variance on k matched samples g08af nagf_nonpar_test_kruskalKruskal–Wallis one-way analysis of variance on k samples of unequal size g08ag nagf_nonpar_test_wilcoxonPerforms the Wilcoxon one-sample (matched pairs) signed rank test g08ah nagf_nonpar_test_mwuPerforms the Mann–Whitney U test on two independent samples g08aj nagf_nonpar_prob_mwu_notiesComputes the exact probabilities for the Mann–Whitney U statistic, no ties in pooled sample g08ak nagf_nonpar_prob_mwu_tiesComputes the exact probabilities for the Mann–Whitney U statistic, ties in pooled sample g08al nagf_nonpar_test_cochranqPerforms the Cochran Q test on cross-classified binary data g08ba nagf_nonpar_test_mooddavidMood's and David's tests on two samples of unequal size g08cb nagf_nonpar_test_ks_1samplePerforms the one-sample Kolmogorov–Smirnov test for standard distributions g08cc nagf_nonpar_test_ks_1sample_userPerforms the one-sample Kolmogorov–Smirnov test for a user-supplied distribution g08cd nagf_nonpar_test_ks_2samplePerforms the two-sample Kolmogorov–Smirnov test g08cg nagf_nonpar_test_chisqPerforms the χ2 goodness-of-fit test, for standard continuous distributions g08ch nagf_nonpar_gofstat_anddarCalculates the Anderson–Darling goodness-of-fit test statistic g08cj nagf_nonpar_gofstat_anddar_unifCalculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of uniformly distributed data g08ck nagf_nonpar_gofstat_anddar_normalCalculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of a fully-unspecified Normal distribution g08cl nagf_nonpar_gofstat_anddar_expCalculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of an unspecified exponential distribution g08da nagf_nonpar_concordance_kendallKendall's coefficient of concordance g08ea nagf_nonpar_randtest_runsPerforms the runs up or runs down test for randomness g08eb nagf_nonpar_randtest_pairsPerforms the pairs (serial) test for randomness g08ec nagf_nonpar_randtest_tripletsPerforms the triplets test for randomness g08ed nagf_nonpar_randtest_gapsPerforms the gaps test for randomness g08ra nagf_nonpar_rank_regsnRegression using ranks, uncensored data g08rb nagf_nonpar_rank_regsn_censoredRegression using ranks, right-censored data
Back to G08 Chapter Summary

### G10 – Smoothing in Statistics

Examples of routines and methods in this chapter:

 g10ab nagf_smooth_fit_splineFit cubic smoothing spline, smoothing parameter given g10ac nagf_smooth_fit_spline_parestFit cubic smoothing spline, smoothing parameter estimated g10bb nagf_smooth_kerndens_gaussKernel density estimate using Gaussian kernel (thread safe) g10ca nagf_smooth_data_runningmedianCompute smoothed data sequence using running median smoothers g10za nagf_smooth_data_orderReorder data to give ordered distinct observations
Back to G10 Chapter Summary

### G11 – Contingency Table Analysis

Examples of routines and methods in this chapter:

 g11aa nagf_contab_chisqχ2 statistics for two-way contingency table g11ba nagf_contab_tabulate_statComputes multiway table from set of classification factors using selected statistic g11bb nagf_contab_tabulate_percentileComputes multiway table from set of classification factors using given percentile/quantile g11bc nagf_contab_tabulate_marginComputes marginal tables for multiway table computed by g11ba or g11bb g11ca nagf_contab_condl_logisticReturns parameter estimates for the conditional analysis of stratified data g11sa nagf_contab_binaryContingency table, latent variable model for binary data g11sb nagf_contab_binary_serviceFrequency count for g11sa
Back to G11 Chapter Summary

### G12 – Survival Analysis

Examples of routines and methods in this chapter:

 g12aa nagf_surviv_kaplanmeierComputes Kaplan–Meier (product-limit) estimates of survival probabilities g12ab nagf_surviv_logrankComputes rank statistics for comparing survival curves g12ba nagf_surviv_coxmodelFits Cox's proportional hazard model g12za nagf_surviv_coxmodel_risksetsCreates the risk sets associated with the Cox proportional hazards model for fixed covariates
Back to G12 Chapter Summary

### G13 – Time Series Analysis

Examples of routines and methods in this chapter:

 g13aa nagf_tsa_uni_diffUnivariate time series, seasonal and non-seasonal differencing g13ab nagf_tsa_uni_autocorrUnivariate time series, sample autocorrelation function g13ac nagf_tsa_uni_autocorr_partUnivariate time series, partial autocorrelations from autocorrelations g13ad nagf_tsa_uni_arima_prelimUnivariate time series, preliminary estimation, seasonal ARIMA model g13ae nagf_tsa_uni_arima_estimUnivariate time series, estimation, seasonal ARIMA model (comprehensive) g13af nagf_tsa_uni_arima_estim_easyUnivariate time series, estimation, seasonal ARIMA model (easy-to-use) g13ag nagf_tsa_uni_arima_updateUnivariate time series, update state set for forecasting g13ah nagf_tsa_uni_arima_forecast_stateUnivariate time series, forecasting from state set g13aj nagf_tsa_uni_arima_forcecastUnivariate time series, state set and forecasts, from fully specified seasonal ARIMA model g13am nagf_tsa_uni_smooth_expUnivariate time series, exponential smoothing g13as nagf_tsa_uni_arima_residUnivariate time series, diagnostic checking of residuals, following g13ae or g13af g13au nagf_tsa_uni_meansComputes quantities needed for range-mean or standard deviation-mean plot g13aw nagf_tsa_uni_dickey_fuller_unitComputes (augmented) Dickey–Fuller unit root test statistic g13ba nagf_tsa_multi_filter_arimaMultivariate time series, filtering (pre-whitening) by an ARIMA model g13bb nagf_tsa_multi_filter_transfMultivariate time series, filtering by a transfer function model g13bc nagf_tsa_multi_xcorrMultivariate time series, cross-correlations g13bd nagf_tsa_multi_transf_prelimMultivariate time series, preliminary estimation of transfer function model g13be nagf_tsa_multi_inputmod_estimMultivariate time series, estimation of multi-input model g13bg nagf_tsa_multi_inputmod_updateMultivariate time series, update state set for forecasting from multi-input model g13bh nagf_tsa_multi_inputmod_forecast_stateMultivariate time series, forecasting from state set of multi-input model g13bj nagf_tsa_multi_inputmod_forecastMultivariate time series, state set and forecasts from fully specified multi-input model g13ca nagf_tsa_uni_spectrum_lagUnivariate time series, smoothed sample spectrum using rectangular, Bartlett, Tukey or Parzen lag window g13cb nagf_tsa_uni_spectrum_daniellUnivariate time series, smoothed sample spectrum using spectral smoothing by the trapezium frequency (Daniell) window g13cc nagf_tsa_multi_spectrum_lagMultivariate time series, smoothed sample cross spectrum using rectangular, Bartlett, Tukey or Parzen lag window g13cd nagf_tsa_multi_spectrum_daniellMultivariate time series, smoothed sample cross spectrum using spectral smoothing by the trapezium frequency (Daniell) window g13ce nagf_tsa_multi_spectrum_bivarMultivariate time series, cross amplitude spectrum, squared coherency, bounds, univariate and bivariate (cross) spectra g13cf nagf_tsa_multi_gain_bivarMultivariate time series, gain, phase, bounds, univariate and bivariate (cross) spectra g13cg nagf_tsa_multi_noise_bivarMultivariate time series, noise spectrum, bounds, impulse response function and its standard error g13db nagf_tsa_multi_autocorr_partMultivariate time series, multiple squared partial autocorrelations g13dd nagf_tsa_multi_varma_estimateMultivariate time series, estimation of VARMA model g13dj nagf_tsa_multi_varma_forecastMultivariate time series, forecasts and their standard errors g13dk nagf_tsa_multi_varma_updateMultivariate time series, updates forecasts and their standard errors g13dl nagf_tsa_multi_diffMultivariate time series, differences and/or transforms g13dm nagf_tsa_multi_corrmat_crossMultivariate time series, sample cross-correlation or cross-covariance matrices g13dn nagf_tsa_multi_corrmat_partlagMultivariate time series, sample partial lag correlation matrices, χ2 statistics and significance levels g13dp nagf_tsa_multi_regmat_partialMultivariate time series, partial autoregression matrices g13ds nagf_tsa_multi_varma_diagMultivariate time series, diagnostic checking of residuals, following g13dd g13dx nagf_tsa_uni_arma_rootsCalculates the zeros of a vector autoregressive (or moving average) operator g13ea nagf_tsa_multi_kalman_sqrt_varCombined measurement and time update, one iteration of Kalman filter, time-varying, square root covariance filter g13eb nagf_tsa_multi_kalman_sqrt_invarCombined measurement and time update, one iteration of Kalman filter, time-invariant, square root covariance filter g13ej nagf_tsa_kalman_unscented_state_revcomCombined time and measurement update, one iteration of the Unscented Kalman Filter for a nonlinear state space model, with additive noise (reverse communication) g13ek nagf_tsa_kalman_unscented_stateCombined time and measurement update, one iteration of the Unscented Kalman Filter for a nonlinear state space model, with additive noise g13fa nagf_tsa_uni_garch_asym1_estimUnivariate time series, parameter estimation for either a symmetric GARCH process or a GARCH process with asymmetry of the form (εt-1+γ)2 g13fb nagf_tsa_uni_garch_asym1_forecastUnivariate time series, forecast function for either a symmetric GARCH process or a GARCH process with asymmetry of the form (εt-1+γ)2 g13fc nagf_tsa_uni_garch_asym2_estimUnivariate time series, parameter estimation for a GARCH process with asymmetry of the form (|εt-1|+γ⁢εt-1)2 g13fd nagf_tsa_uni_garch_asym2_forecastUnivariate time series, forecast function for a GARCH process with asymmetry of the form (|εt-1|+γ⁢εt-1)2 g13fe nagf_tsa_uni_garch_gjr_estimUnivariate time series, parameter estimation for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process g13ff nagf_tsa_uni_garch_gjr_forecastUnivariate time series, forecast function for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process g13fg nagf_tsa_uni_garch_exp_estimUnivariate time series, parameter estimation for an exponential GARCH (EGARCH) process g13fh nagf_tsa_uni_garch_exp_forecastUnivariate time series, forecast function for an exponential GARCH (EGARCH) process g13me nagf_tsa_inhom_iemaComputes the iterated exponential moving average for a univariate inhomogeneous time series g13mf nagf_tsa_inhom_iema_allComputes the iterated exponential moving average for a univariate inhomogeneous time series, intermediate results are also returned g13mg nagf_tsa_inhom_maComputes the exponential moving average for a univariate inhomogeneous time series g13na nagf_tsa_cp_peltChange point detection, using the PELT algorithm g13nb nagf_tsa_cp_pelt_userChange points detection using the PELT algorithm, user supplied cost function g13nd nagf_tsa_cp_binaryChange point detection, using binary segmentation g13ne nagf_tsa_cp_binary_userChange point detection, using binary segmentation, user supplied cost function
Back to G13 Chapter Summary

### G22 – Linear Model Specification

Examples of routines and methods in this chapter:

 g22ya nagf_blgm_lm_formulaSpecify a linear model via a formula string g22yb nagf_blgm_lm_describe_dataDescribe a dataset g22yc nagf_blgm_lm_design_matrixConstruct a design matrix from a linear model specified using g22ya g22yd nagf_blgm_lm_submodelConstruct a vector indicating which columns of a design matrix to include in a submodel specified using g22ya g22za nagf_blgm_handle_freeDestroy a G22 handle and deallocate all the memory used g22zm nagf_blgm_optsetOption setting routine for Chapter G22 g22zn nagf_blgm_optgetOption getting routine for Chapter G22
Back to G22 Chapter Summary

### H – Operations Research

Examples of routines and methods in this chapter:

 h02bb nagf_mip_ilp_denseInteger LP problem (dense) h02bf nagf_mip_ilp_mpsxInterpret MPSX data file defining IP or LP problem, optimize and print solution h02bu nagf_mip_ilp_mpsx_convertConvert MPSX data file defining IP or LP problem to format required by h02bb or e04mf h02bv nagf_mip_ilp_printPrint IP or LP solutions with user-specified names for rows and columns h02bz nagf_mip_ilp_infoInteger programming solution, supplies further information on solution obtained by h02bb h02cb nagf_mip_iqp_denseInteger QP problem (dense) h02cb nagf_mip_iqp_dense_dummy_monitdummy h02cc nagf_mip_iqp_dense_optfileRead optional parameter values for h02cb from external file h02cd nagf_mip_iqp_dense_optstrSupply optional parameter values to h02cb h02ce nagf_mip_iqp_sparseInteger LP or QP problem (sparse), using e04nk h02ce nagf_mip_iqp_sparse_dummy_monitdummy h02cf nagf_mip_iqp_sparse_optfileRead optional parameter values for h02ce from external file h02cg nagf_mip_iqp_sparse_optstrSupply optional parameter values to h02ce h02da nagf_mip_sqpMixed integer nonlinear programming h02zk nagf_mip_optsetOption setting routine for h02da h02zl nagf_mip_optgetOption getting routine for h02da h03ab nagf_mip_transportationTransportation problem, modified 'stepping stone' method h03ad nagf_mip_shortestpathShortest path problem, Dijkstra's algorithm h03bb nagf_mip_tsp_simannTravelling Salesman Problem, simulated annealing h05aa nagf_mip_best_subset_given_size_revcommBest n subsets of size p (reverse communication) h05ab nagf_mip_best_subset_given_sizeBest n subsets of size p (direct communication)
Back to H Chapter Summary

### J10 –

Examples of routines and methods in this chapter:

Back to J10 Chapter Summary

### M01 – Sorting and Searching

Examples of routines and methods in this chapter:

 m01ca nagf_sort_realvec_sortSort a vector, real numbers m01cb nagf_sort_intvec_sortSort a vector, integer numbers m01cc nagf_sort_charvec_sortSort a vector, character data m01da nagf_sort_realvec_rankRank a vector, real numbers m01db nagf_sort_intvec_rankRank a vector, integer numbers m01dc nagf_sort_charvec_rankRank a vector, character data m01de nagf_sort_realmat_rank_rowsRank rows of a matrix, real numbers m01df nagf_sort_intmat_rank_rowsRank rows of a matrix, integer numbers m01dj nagf_sort_realmat_rank_columnsRank columns of a matrix, real numbers m01dk nagf_sort_intmat_rank_columnsRank columns of a matrix, integer numbers m01dz nagf_sort_arbitrary_rankRank arbitrary data m01ea nagf_sort_realvec_rank_rearrangeRearrange a vector according to given ranks, real numbers m01eb nagf_sort_intvec_rank_rearrangeRearrange a vector according to given ranks, integer numbers m01ec nagf_sort_charvec_rank_rearrangeRearrange a vector according to given ranks, character data m01ed nagf_sort_cmplxvec_rank_rearrangeRearrange a vector according to given ranks, complex numbers m01na nagf_sort_realvec_searchBinary search in set of real numbers m01nb nagf_sort_intvec_searchBinary search in set of integer numbers m01nc nagf_sort_charvec_searchBinary search in set of character data m01za nagf_sort_permute_invertInvert a permutation m01zb nagf_sort_permute_checkCheck validity of a permutation m01zc nagf_sort_permute_decomposeDecompose a permutation into cycles
Back to M01 Chapter Summary

### P01 – Error Trapping

Examples of routines and methods in this chapter:

Back to P01 Chapter Summary

### S – Approximations of Special Functions

Examples of routines and methods in this chapter:

 s01ba nagf_specfun_log_shiftedln (1+x) s01ea nagf_specfun_exp_complexComplex exponential, ez s07aa nagf_specfun_tantan x s09aa nagf_specfun_arcsinarcsin x s09ab nagf_specfun_arccosarccos x s10aa nagf_specfun_tanhtanh x s10ab nagf_specfun_sinhsinh x s10ac nagf_specfun_coshcosh x s11aa nagf_specfun_arctanharctanh x s11ab nagf_specfun_arcsinharcsinh x s11ac nagf_specfun_arccosharccosh x s13aa nagf_specfun_integral_expExponential integral E1(x) s13ac nagf_specfun_integral_cosCosine integral Ci (x) s13ad nagf_specfun_integral_sinSine integral Si (x) s14aa nagf_specfun_gammaGamma function s14ab nagf_specfun_gamma_log_realLog gamma function, real argument s14ac nagf_specfun_polygammaψ(x)-ln x s14ad nagf_specfun_polygamma_derivScaled derivatives of ψ(x) s14ae nagf_specfun_psi_deriv_realPolygamma function ψ(n)(x) for real x s14af nagf_specfun_psi_deriv_complexPolygamma function ψ(n)(z) for complex z s14ag nagf_specfun_gamma_log_complexLogarithm of the gamma function ln Γ(z), complex argument s14ah nagf_specfun_gamma_log_scaled_realScaled log gamma function s14ba nagf_specfun_gamma_incompleteIncomplete gamma functions P(a,x) and Q(a,x) s14cb nagf_specfun_beta_log_realLogarithm of the beta function ln B(a,b) s14cc nagf_specfun_beta_incompleteRegularized incomplete beta function Ix(a,b) and its complement 1-Ix s15ab nagf_specfun_cdf_normalCumulative Normal distribution function P(x) s15ac nagf_specfun_compcdf_normalComplement of cumulative Normal distribution function Q(x) s15ad nagf_specfun_erfc_realComplement of error function erfc (x) s15ae nagf_specfun_erf_realError function erf (x) s15af nagf_specfun_dawsonDawson's integral s15ag nagf_specfun_erfcx_realScaled complement of error function, erfcx (x) s15dd nagf_specfun_erfc_complexScaled complex complement of error function, exp (-z2)erfc (-iz) s17ac nagf_specfun_bessel_y0_realBessel function Y0(x) s17ad nagf_specfun_bessel_y1_realBessel function Y1(x) s17ae nagf_specfun_bessel_j0_realBessel function J0(x) s17af nagf_specfun_bessel_j1_realBessel function J1(x) s17ag nagf_specfun_airy_ai_realAiry function Ai (x) s17ah nagf_specfun_airy_bi_realAiry function Bi (x) s17aj nagf_specfun_airy_ai_derivAiry function Ai′ (x) s17ak nagf_specfun_airy_bi_derivAiry function Bi′ (x) s17al nagf_specfun_bessel_zerosZeros of Bessel functions Jα(x), Jα′(x), Yα(x) or Yα′(x) s17aq nagf_specfun_bessel_y0_real_vectorBessel function vectorized Y0(x) s17ar nagf_specfun_bessel_y1_real_vectorBessel function vectorized Y1(x) s17as nagf_specfun_bessel_j0_real_vectorBessel function vectorized J0(x) s17at nagf_specfun_bessel_j1_real_vectorBessel function vectorized J1(x) s17au nagf_specfun_airy_ai_real_vectorAiry function vectorized Ai (x) s17av nagf_specfun_airy_bi_real_vectorAiry function vectorized Bi (x) s17aw nagf_specfun_airy_ai_deriv_vectorDerivatives of the Airy function, vectorized Ai′ (x) s17ax nagf_specfun_airy_bi_deriv_vectorDerivatives of the Airy function, vectorized Bi′ (x) s17dc nagf_specfun_bessel_y_complexBessel functions Yν+a(z), real a≥0, complex z, ν=0,1,2,… s17de nagf_specfun_bessel_j_complexBessel functions Jν+a(z), real a≥0, complex z, ν=0,1,2,… s17dg nagf_specfun_airy_ai_complexAiry functions Ai (z) and Ai′ (z), complex z s17dh nagf_specfun_airy_bi_complexAiry functions Bi (z) and Bi′ (z), complex z s17dl nagf_specfun_hankel_complexHankel functions Hν+a(j)(z), j=1,2, real a≥0, complex z, ν=0,1,2,… s17ga nagf_specfun_struve_h0Struve function of order 0, H0(x) s17gb nagf_specfun_struve_h1Struve function of order 1, H1(x) s18ac nagf_specfun_bessel_k0_realModified Bessel function K0(x) s18ad nagf_specfun_bessel_k1_realModified Bessel function K1(x) s18ae nagf_specfun_bessel_i0_realModified Bessel function I0(x) s18af nagf_specfun_bessel_i1_realModified Bessel function I1(x) s18aq nagf_specfun_bessel_k0_real_vectorModified Bessel function vectorized K0(x) s18ar nagf_specfun_bessel_k1_real_vectorModified Bessel function vectorized K1(x) s18as nagf_specfun_bessel_i0_real_vectorModified Bessel function vectorized I0(x) s18at nagf_specfun_bessel_i1_real_vectorModified Bessel function vectorized I1(x) s18cc nagf_specfun_bessel_k0_scaledScaled modified Bessel function exK0(x) s18cd nagf_specfun_bessel_k1_scaledScaled modified Bessel function exK1(x) s18ce nagf_specfun_bessel_i0_scaledScaled modified Bessel function e-|x|I0(x) s18cf nagf_specfun_bessel_i1_scaledScaled modified Bessel function e-|x|I1(x) s18cq nagf_specfun_bessel_k0_scaled_vectorScaled modified Bessel function vectorized exK0(x) s18cr nagf_specfun_bessel_k1_scaled_vectorScaled modified Bessel function vectorized exK1(x) s18cs nagf_specfun_bessel_i0_scaled_vectorScaled modified Bessel function vectorized e-|x|I0(x) s18ct nagf_specfun_bessel_i1_scaled_vectorScaled modified Bessel function vectorized e-|x|I1(x) s18dc nagf_specfun_bessel_k_complexModified Bessel functions Kν+a(z), real a≥0, complex z, ν=0,1,2,… s18de nagf_specfun_bessel_i_complexModified Bessel functions Iν+a(z), real a≥0, complex z, ν=0,1,2,… s18gk nagf_specfun_bessel_j_seq_complexBessel function of the 1st kind Jα±n(z) s18ga nagf_specfun_struve_l0Modified Struve function of order 0, L0(x) s18gb nagf_specfun_struve_l1Modified Struve function of order 1, L1(x) s18gc nagf_specfun_struve_i0ml0The function I0(x)-L0(x), where I0(x) is a modified Bessel function and L0(x) is a Struve function s18gd nagf_specfun_struve_i1ml1The function I1(x)-L1(x), where I1(x) is a modified Bessel function and L1(x) is a Struve function s19aa nagf_specfun_kelvin_berKelvin function ber x s19ab nagf_specfun_kelvin_beiKelvin function bei x s19ac nagf_specfun_kelvin_kerKelvin function ker x s19ad nagf_specfun_kelvin_keiKelvin function kei x s19an nagf_specfun_kelvin_ber_vectorKelvin function vectorized ber x s19ap nagf_specfun_kelvin_bei_vectorKelvin function vectorized bei x s19aq nagf_specfun_kelvin_ker_vectorKelvin function vectorized ker x s19ar nagf_specfun_kelvin_kei_vectorKelvin function vectorized kei x s20ac nagf_specfun_fresnel_sFresnel integral S(x) s20ad nagf_specfun_fresnel_cFresnel integral C(x) s20aq nagf_specfun_fresnel_s_vectorFresnel integral vectorized S(x) s20ar nagf_specfun_fresnel_c_vectorFresnel integral vectorized C(x) s21ba nagf_specfun_ellipint_symm_1_degenDegenerate symmetrised elliptic integral of 1st kind RC(x,y) s21bb nagf_specfun_ellipint_symm_1Symmetrised elliptic integral of 1st kind RF(x,y,z) s21bc nagf_specfun_ellipint_symm_2Symmetrised elliptic integral of 2nd kind RD(x,y,z) s21bd nagf_specfun_ellipint_symm_3Symmetrised elliptic integral of 3rd kind RJ(x,y,z,r) s21be nagf_specfun_ellipint_legendre_1Elliptic integral of 1st kind, Legendre form, F(ϕ∣m) s21bf nagf_specfun_ellipint_legendre_2Elliptic integral of 2nd kind, Legendre form, E(ϕ∣m) s21bg nagf_specfun_ellipint_legendre_3Elliptic integral of 3rd kind, Legendre form, Π(n;ϕ∣m) s21bh nagf_specfun_ellipint_complete_1Complete elliptic integral of 1st kind, Legendre form, K(m) s21bj nagf_specfun_ellipint_complete_2Complete elliptic integral of 2nd kind, Legendre form, E(m) s21ca nagf_specfun_jacellip_realJacobian elliptic functions sn, cn and dn of real argument s21cb nagf_specfun_jacellip_complexJacobian elliptic functions sn, cn and dn of complex argument s21cc nagf_specfun_jactheta_realJacobian theta functions θk(x,q) of real argument s21da nagf_specfun_ellipint_general_2General elliptic integral of 2nd kind F(z,k′,a,b) of complex argument s22aa nagf_specfun_legendre_pLegendre functions of 1st kind Pnm(x) or Pnm-(x) s22ba nagf_specfun_1f1_realReal confluent hypergeometric function 1F1(a;b;x) s22bb nagf_specfun_1f1_real_scaledReal confluent hypergeometric function 1F1(a;b;x) in scaled form s22be nagf_specfun_2f1_realReal Gauss hypergeometric function 2F1(a,b;c;x) s22bf nagf_specfun_2f1_real_scaledReal Gauss hypergeometric function 2F1(a,b;c;x) in scaled form s30aa nagf_specfun_opt_bsm_priceBlack–Scholes–Merton option pricing formula s30ab nagf_specfun_opt_bsm_greeksBlack–Scholes–Merton option pricing formula with Greeks s30ba nagf_specfun_opt_lookback_fls_priceFloating-strike lookback option pricing formula in the Black-Scholes-Merton model s30bb nagf_specfun_opt_lookback_fls_greeksFloating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model s30ca nagf_specfun_opt_binary_con_priceBinary option, cash-or-nothing pricing formula s30cb nagf_specfun_opt_binary_con_greeksBinary option, cash-or-nothing pricing formula with Greeks s30cc nagf_specfun_opt_binary_aon_priceBinary option, asset-or-nothing pricing formula s30cd nagf_specfun_opt_binary_aon_greeksBinary option, asset-or-nothing pricing formula with Greeks s30fa nagf_specfun_opt_barrier_std_priceStandard barrier option pricing formula s30ja nagf_specfun_opt_jumpdiff_merton_priceJump-diffusion, Merton's model, option pricing formula s30jb nagf_specfun_opt_jumpdiff_merton_greeksJump-diffusion, Merton's model, option pricing formula with Greeks s30na nagf_specfun_opt_heston_priceHeston's model option pricing formula s30nb nagf_specfun_opt_heston_greeksHeston's model option pricing formula with Greeks s30nc nagf_specfun_opt_heston_termHeston's model option pricing with term structure s30qc nagf_specfun_opt_amer_bs_priceAmerican option, Bjerksund and Stensland pricing formula s30sa nagf_specfun_opt_asian_geom_priceAsian option, geometric continuous average rate pricing formula s30sb nagf_specfun_opt_asian_geom_greeksAsian option, geometric continuous average rate pricing formula with Greeks
Back to S Chapter Summary

### X01 – Mathematical Constants

Examples of routines and methods in this chapter:

 x01aa nagf_math_piProvides the mathematical constant π x01ab nagf_math_eulerProvides the mathematical constant γ (Euler's constant)
Back to X01 Chapter Summary

### X02 – Machine Constants

Examples of routines and methods in this chapter:

 x02ah nagf_machine_sinarg_maxThe largest permissible argument for sin and cos x02aj nagf_machine_precisionThe machine precision x02ak nagf_machine_real_smallestThe smallest positive model number x02al nagf_machine_real_largestThe largest positive model number x02am nagf_machine_real_safeThe safe range parameter x02an nagf_machine_complex_safeThe safe range parameter for complex floating-point arithmetic x02bb nagf_machine_integer_maxThe largest representable integer x02be nagf_machine_decimal_digitsThe maximum number of decimal digits that can be represented x02bh nagf_machine_model_baseThe floating-point model parameter, b x02bj nagf_machine_model_digitsThe floating-point model parameter, p x02bk nagf_machine_model_minexpThe floating-point model parameter emin x02bl nagf_machine_model_maxexpThe floating-point model parameter emax
Back to X02 Chapter Summary

### X03 – Inner Products

Examples of routines and methods in this chapter:

Back to X03 Chapter Summary

### X04 – Input/Output Utilities

Examples of routines and methods in this chapter:

 x04aa nagf_file_set_unit_errorReturn or set unit number for error messages x04ab nagf_file_set_unit_advisoryReturn or set unit number for advisory messages x04ac nagf_file_openOpen unit number for reading, writing or appending, and associate unit with named file x04ad nagf_file_closeClose file associated with given unit number x04ba nagf_file_line_writeWrite formatted record to external file x04bb nagf_file_line_readRead formatted record from external file x04ca nagf_file_print_matrix_real_genPrint real general matrix (easy-to-use) x04cb nagf_file_print_matrix_real_gen_compPrint real general matrix (comprehensive) x04cc nagf_file_print_matrix_real_packedPrint real packed triangular matrix (easy-to-use) x04cd nagf_file_print_matrix_real_packed_compPrint real packed triangular matrix (comprehensive) x04ce nagf_file_print_matrix_real_bandPrint real packed banded matrix (easy-to-use) x04cf nagf_file_print_matrix_real_band_compPrint real packed banded matrix (comprehensive) x04da nagf_file_print_matrix_complex_genPrint complex general matrix (easy-to-use) x04db nagf_file_print_matrix_complex_gen_compPrint complex general matrix (comprehensive) x04dc nagf_file_print_matrix_complex_packedPrint complex packed triangular matrix (easy-to-use) x04dd nagf_file_print_matrix_complex_packed_compPrint complex packed triangular matrix (comprehensive) x04de nagf_file_print_matrix_complex_bandPrint complex packed banded matrix (easy-to-use) x04df nagf_file_print_matrix_complex_band_compPrint complex packed banded matrix (comprehensive) x04ea nagf_file_print_matrix_integerPrint integer matrix (easy-to-use) x04eb nagf_file_print_matrix_integer_compPrint integer matrix (comprehensive)
Back to X04 Chapter Summary

### X05 – Date and Time Utilities

Examples of routines and methods in this chapter:

 x05aa nagf_time_date_arrayReturn date and time as an array of integers x05ab nagf_time_date_array_stringConvert array of integers representing date and time to character string x05ac nagf_time_date_string_compareCompare two character strings representing date and time x05ba nagf_time_cpuReturn the CPU time
Back to X05 Chapter Summary

### X06 – OpenMP Utilities

Examples of routines and methods in this chapter:

Back to X06 Chapter Summary

### X07 – IEEE Arithmetic

Examples of routines and methods in this chapter:

 x07aa nagf_ieee_is_finiteDetermines whether its argument has a finite value x07ab nagf_ieee_is_nanDetermines whether its argument is a NaN (Not A Number) x07ba nagf_ieee_create_infinityCreates a signed infinite value x07bb nagf_ieee_create_nanCreates a NaN (Not A Number) x07ca nagf_ieee_get_exception_modeGets current behaviour of floating-point exceptions x07cb nagf_ieee_set_exception_modeSets behaviour of floating-point exceptions
Back to X07 Chapter Summary

### X10 – Automatic Differentiation Utilities

Examples of routines and methods in this chapter:

 x10aa Create a configuration data object x10ab Remove a configuration data object x10ac Set the algorithmic mode x10ad Get the algorithmic mode x10ba Create a callback data object x10bb Insert the location of the AD callback into the callback data object x10bc Set the callback algorithmic mode x10bd Get the callback algorithmic mode x10be Write an integer (working precision) scalar to the callback data object x10bf Write a real (working precision) scalar to the callback data object x10bj Write a scalar of type nagad_a1w_w_rtype to the callback data object x10ce Read an integer scalar from the callback data object x10cf Read a real (working precision) scalar from the callback data object x10cj Read a scalar of type nagad_a1w_w_rtype from the callback data object
Back to X10 Chapter Summary

© The Numerical Algorithms Group Ltd, Oxford, UK. 2018