NAG C Library News, Mark 23

This new release of the NAG C Library has been given the designation Mark 23 whereas the previous release was called Mark 9. This non-sequential numbering reflects the fact that, with the inclusion of this substantial update, the functionality provided in this release of the library is broadly comparable with that available in other NAG Libraries at that Mark.

At Mark 23 of the NAG C Library new functionality has been introduced in addition to improvements in existing areas. The Library now contains 1452 user-callable functions, all of which are documented, of which 345 are new at this mark.

Two new chapters have been introduced and there have been extensions in functionality included in the areas of statistics, nonlinear equations, wavelets, ordinary differential equations, interpolation, surface fitting, optimization, matrix operations, linear algebra, large scale linear systems, and special functions.

Chapter c05 (Roots of One or More Transcendental Equations) has a new function for solving sparse nonlinear systems and a new function for determining values of the complex Lambert–W function. Additionally, some functions have been added to provide more breadth to zero finding and solving dense nonlinear systems.

Chapter c06 (Fourier Transforms) has a function for summing a Chebyshev series at a vector of points.

Chapter c09 (Wavelet Transforms) has added one-dimensional continuous and two-dimensional discrete wavelet transform functions.

Chapter d01 (Quadrature) has additional functions for multidimensional integrals, which include integration over the $n$-hypersphere, a number theoretic method and Gaussian rules.

Chapter d02 (Ordinary Differential Equations) has a new suite of functions for solving boundary-value problems by an implementation of the Chebyshev pseudospectral method.

A new Chapter d04 chapter (Numerical Differentiation) provides functions to enable you to estimate derivatives of order up to $14$.

Linear non-singular Fredholm integral equations of the second kind may now be solved using functions from the new Chapter d05 (Integral Equations). In addition the chapter contains functions for second kind, nonlinear Volterra convolution equations and both first and second kind, weakly singular, nonlinear convolution Volterra–Abel equations.

Chapter e01 (Interpolation) adds a modified Shepard's method for multivariate scattered data in four and five dimensions. It also includes the one-dimensional methods of Aitken and Everett.

Chapter e02 (Curve and Surface Fitting) has evaluation of the derivatives of spline surfaces.

Chapter e04 (Minimizing or Maximizing a Function) has a new function to minimize an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) using a sequential quadratic programming (SQP) method using reverse communication. A function to minimize a smooth function and several variables subject to upper and lower bounds on the variable, using methods of quadratic approximation based on BOBYQA has also been introduced.

Chapter e05 (Global Optimization of a Function) has new functions implementing Particle Swarm Optimization and multi-start Sequential Quadratic Programming. The existing function for multi-level coordinate search now allows equality bound constraints.

Chapter f01 (Matrix Operations, Including Inversion) has new routines for functions of real and complex matrices, including matrix log and exponentials, functions of symmetric/Hermitian matrices and functions of general matrices using user-supplied derivatives.

Chapter f07 (Linear Equations (LAPACK)) has LAPACK 3.2 simple and expert drivers for the solution systems of linear equations, where the system is general (full, banded or tridiagonal), symmetric/Hermitian positive definite (full with full or packed storage, banded or tridiagonal), and symmetric/Hermitian (full or packed storage). Additionally there are equilibration functions for various system types and support routines for tridiagonal systems.

Chapter f08 (Least Squares and Eigenvalue Problems (LAPACK)) has LAPACK functions for: solving under and over-determined systems of linear equations; obtaining minimum-norm solutions by SVD or divide-and-conquer algorithms; computing the QR, QL, RQ, RZ factorizations of real and complex matrices (and subsequently forming or applying Q or Z); find some or all eigenvalues/vectors of symmetric/Hermitian matrices (full, packed, banded or tridiagonal); computing the SVD of general matrices using the QR or divide-and-conquer algorithms; computing the Schur form for square matrices. Additionally, for matrix pairs, there are functions for computing the generalized eigenvalues, generalized SVD and Schur forms, and generalized QR and RQ factorizations.

Chapter f11 (Large Scale Eigenproblems) has new suites of reverse communication functions for the iterative solution of sparse symmetric/Hermitian and nonsymmetric/non-Hermitian systems of linear equations; black-box routines for solving sparse Hermitian/non-Hermitian complex systems employing a variety of methods and preconditioners; and some sparse matrix utility functions.

Chapter g01 (Simple Calculations on Statistical Data) has new functions for quantiles of streamed data, bivariate Student's $t$-distribution and two probability density functions.

Chapter g02 (Correlation and Regression Analysis) has new functions for nearest correlation matrices, hierarchical mixed effects regression, and quantile regression.

Chapter g05 (Random Number Generators) has a new function for skip-ahead by powers of $2$ and weighted sampling without replacement. In addition, the suite of base generators has been extended to include the L'Ecuyer MRG32k3a generator. Skip-ahead for the Mersenne Twister base generator is also now available.

Chapter g07 (Univariate Estimation) has new functions for Pareto distribution parameter estimation and outlier detection by the method of Peirce.

Chapter g08 (Nonparametric Statistics) has routines for the Anderson–Darling goodness-of-fit test.

Chapter g12 (Survival Analysis) has a new function for computing rank statistics when comparing survival curves.

Chapter s (Approximations of Special Functions) has new beta and incomplete beta functions and the S30 sub-chapter has a new function for computing Greeks for Heston's model option pricing formula.

In addition there have been a number of enhancements included in Chapters g01 and s. These do not provide new functionality but instead focus on usability. Typically simple functions in these chapters return a single numeric value; a number of these have now been complemented by equivalent functions that conveniently return an array of such values via a single call.

The 345 new user-callable functions included in the NAG C Library at Mark 23 are as follows.

FunctionName |
Purpose |

c05auc | Zero of continuous function, Brent algorithm, from a given starting value, binary search for interval |

c05awc | Zero of continuous function, continuation method, from a given starting value |

c05ayc | Zero of continuous function in a given interval, Brent algorithm |

c05bbc | Values of Lambert's $W$ function, $W\left(z\right)$ |

c05qbc | Solution of a system of nonlinear equations using function values only (easy-to-use) |

c05qcc | Solution of a system of nonlinear equations using function values only (comprehensive) |

c05qdc | Solution of a system of nonlinear equations using function values only (reverse communication) |

c05qsc | Solution of a sparse system of nonlinear equations using function values only (easy-to-use) |

c05rbc | Solution of a system of nonlinear equations using first derivatives (easy-to-use) |

c05rcc | Solution of a system of nonlinear equations using first derivatives (comprehensive) |

c05rdc | Solution of a system of nonlinear equations using first derivatives (reverse communication) |

c05zdc | Check user's function for calculating first derivatives of a set of nonlinear functions of several variables |

c06dcc | Sum of a Chebyshev series at a set of points |

c09abc | Two-dimensional wavelet filter initialization |

c09bac | One-dimensional real continuous wavelet transform |

c09eac | Two-dimensional discrete wavelet transform |

c09ebc | Two-dimensional inverse discrete wavelet transform |

c09ecc | Two-dimensional multi-level discrete wavelet transform |

c09edc | Two-dimensional inverse multi-level discrete wavelet transform |

d01bdc | One-dimensional quadrature, non-adaptive, finite interval |

d01dac | Two-dimensional quadrature, finite region |

d01fbc | Multidimensional Gaussian quadrature over hyper-rectangle |

d01fdc | Multidimensional quadrature, Sag–Szekeres method, general product region or $n$-sphere |

d01gdc | Multidimensional quadrature, general product region, number-theoretic method |

d01gyc | Korobov optimal coefficients for use in nag_quad_md_numth_vec (d01gdc), when number of points is prime |

d01gzc | Korobov optimal coefficients for use in nag_quad_md_numth_vec (d01gdc), when number of points is product of two primes |

d01pac | Multidimensional quadrature over an $n$-simplex |

d01tbc | Pre-computed weights and abscissae for Gaussian quadrature rules, restricted choice of rule |

d01tcc | Calculation of weights and abscissae for Gaussian quadrature rules, general choice of rule |

d02uac | Coefficients of Chebyshev interpolating polynomial from function values on Chebyshev grid |

d02ubc | Function or low-order-derivative values on Chebyshev grid from coefficients of Chebyshev interpolating polynomial |

d02ucc | Chebyshev Gauss–Lobatto grid generation |

d02udc | Differentiate a function by the FFT using function values on Chebyshev grid |

d02uec | Solve linear constant coefficient boundary value problem on Chebyshev grid, Integral formulation |

d02uwc | Interpolate a function from Chebyshev grid to uniform grid using barycentric Lagrange interpolation |

d02uyc | Clenshaw–Curtis quadrature weights for integration using computed Chebyshev coefficients |

d02uzc | Chebyshev polynomial evaluation, ${T}_{k}\left(x\right)$ |

d04aac | Numerical differentiation, derivatives up to order 14, function of one real variable |

d04bac | Numerical differentiation, user-supplied function values, derivatives up to order $14$, derivatives with respect to one real variable |

d04bbc | Generates sample points for function evaluations by nag_numdiff_1d_real_eval (d04bac) |

d05aac | Linear non-singular Fredholm integral equation, second kind, split kernel |

d05abc | Linear non-singular Fredholm integral equation, second kind, smooth kernel |

d05bac | Nonlinear Volterra convolution equation, second kind |

d05bdc | Nonlinear convolution Volterra–Abel equation, second kind, weakly singular |

d05bec | Nonlinear convolution Volterra–Abel equation, first kind, weakly singular |

d05bwc | Generate weights for use in solving Volterra equations |

d05byc | Generate weights for use in solving weakly singular Abel-type equations |

e01aac | Interpolated values, Aitken's technique, unequally spaced data, one variable |

e01abc | Interpolated values, Everett's formula, equally spaced data, one variable |

e01tkc | Interpolating functions, modified Shepard's method, four variables |

e01tlc | Interpolated values, evaluate interpolant computed by nag_4d_shep_interp (e01tkc), function and first derivatives, four variables |

e01tmc | Interpolating functions, modified Shepard's method, five variables |

e01tnc | Interpolated values, evaluate interpolant computed by nag_5d_shep_interp (e01tmc), function and first derivatives, five variables |

e02dhc | Evaluation of spline surface at mesh of points with derivatives |

e04jcc | Minimum by quadratic approximation, function of several variables, simple bounds, using function values only |

e04udc | Supply optional argument values for nag_opt_nlp (e04ucc) or nag_opt_nlp_revcomm (e04ufc) from external file |

e04uec | Supply optional argument values to nag_opt_nlp (e04ucc) or nag_opt_nlp_revcomm (e04ufc) |

e04ufc | Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (reverse communication, comprehensive) |

e04wbc | Initialization function for nag_opt_nlp_revcomm (e04ufc) |

e05sac | Global optimization using particle swarm algorithm (PSO), bound constraints only |

e05sbc | Global optimization using particle swarm algorithm (PSO), comprehensive |

e05ucc | Global optimization using multi-start, nonlinear constraints |

e05zkc | Option setting routine for nag_glopt_bnd_pso (e05sac), nag_glopt_nlp_pso (e05sbc) and nag_glopt_nlp_multistart_sqp (e05ucc) |

e05zlc | Option getting routine for nag_glopt_bnd_pso (e05sac), nag_glopt_nlp_pso (e05sbc) and nag_glopt_nlp_multistart_sqp (e05ucc) |

f01efc | Function of a real symmetric matrix |

f01ejc | Real matrix logarithm |

f01ekc | Exponential, sine, cosine, sinh or cosh of a real matrix (Schur–Parlett algorithm) |

f01emc | Function of a real matrix (using user-supplied derivatives) |

f01fcc | Complex matrix exponential |

f01fdc | Complex Hermitian matrix exponential |

f01ffc | Function of a complex Hermitian matrix |

f01fjc | Complex matrix logarithm |

f01fkc | Exponential, sine, cosine, sinh or cosh of a complex matrix (Schur–Parlett algorithm) |

f01fmc | Function of a complex matrix (using user-supplied derivatives) |

f03bac | $LU$ factorization and determinant of real matrix |

f03bfc | $L{L}^{\mathrm{T}}$ factorization and determinant of real symmetric positive definite matrix |

f03bhc | Determinant of real symmetric positive definite banded matrix |

f03bnc | Determinant of complex matrix |

f07aac | Computes the solution to a real system of linear equations |

f07abc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real system of linear equations |

f07acc | Mixed precision real system solver |

f07afc | Computes row and column scalings intended to equilibrate a general real matrix and reduce its condition number |

f07anc | Computes the solution to a complex system of linear equations |

f07apc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex system of linear equations |

f07aqc | Mixed precision complex system solver |

f07atc | Computes row and column scalings intended to equilibrate a general complex matrix and reduce its condition number |

f07bac | Computes the solution to a real banded system of linear equations |

f07bbc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real banded system of linear equations |

f07bfc | Computes row and column scalings intended to equilibrate a real banded matrix and reduce its condition number |

f07bnc | Computes the solution to a complex banded system of linear equations |

f07bpc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex banded system of linear equations |

f07btc | Computes row and column scalings intended to equilibrate a complex banded matrix and reduce its condition number |

f07cac | Computes the solution to a real tridiagonal system of linear equations |

f07cbc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real tridiagonal system of linear equations |

f07cdc | $LU$ factorization of real tridiagonal matrix |

f07cec | Solves a real tridiagonal system of linear equations using the $LU$ factorization computed by nag_dgttrf (f07cdc) |

f07cgc | Estimates the reciprocal of the condition number of a real tridiagonal matrix using the $LU$ factorization computed by nag_dgttrf (f07cdc) |

f07chc | Refined solution with error bounds of real tridiagonal system of linear equations, multiple right-hand sides |

f07cnc | Computes the solution to a complex tridiagonal system of linear equations |

f07cpc | Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex tridiagonal system of linear equations |

f07crc | $LU$ factorization of complex tridiagonal matrix |

f07csc | Solves a complex tridiagonal system of linear equations using the $LU$ factorization computed by nag_dgttrf (f07cdc) |

f07cuc | Estimates the reciprocal of the condition number of a complex tridiagonal matrix using the $LU$ factorization computed by nag_dgttrf (f07cdc) |

f07cvc | Refined solution with error bounds of complex tridiagonal system of linear equations, multiple right-hand sides |

f07fac | Computes the solution to a real symmetric positive definite system of linear equations |

f07fbc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite system of linear equations |

f07ffc | Computes row and column scalings intended to equilibrate a real symmetric positive definite matrix and reduce its condition number |

f07fnc | Computes the solution to a complex Hermitian positive definite system of linear equations |

f07fpc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite system of linear equations |

f07ftc | Computes row and column scalings intended to equilibrate a complex Hermitian positive definite matrix and reduce its condition number |

f07gac | Computes the solution to a real symmetric positive definite system of linear equations, packed storage |

f07gbc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite system of linear equations, packed storage |

f07gfc | Computes row and column scalings intended to equilibrate a real symmetric positive definite matrix and reduce its condition number, packed storage |

f07gnc | Computes the solution to a complex Hermitian positive definite system of linear equations, packed storage |

f07gpc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite system of linear equations, packed storage |

f07gtc | Computes row and column scalings intended to equilibrate a complex Hermitian positive definite matrix and reduce its condition number, packed storage |

f07hac | Computes the solution to a real symmetric positive definite banded system of linear equations |

f07hbc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite banded system of linear equations |

f07hfc | Computes row and column scalings intended to equilibrate a real symmetric positive definite banded matrix and reduce its condition number |

f07hnc | Computes the solution to a complex Hermitian positive definite banded system of linear equations |

f07hpc | Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite banded system of linear equations |

f07htc | Computes row and column scalings intended to equilibrate a complex Hermitian positive definite banded matrix and reduce its condition number |

f07jac | Computes the solution to a real symmetric positive definite tridiagonal system of linear equations |

f07jbc | Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite tridiagonal system of linear equations |

f07jdc | Computes the modified Cholesky factorization of a real symmetric positive definite tridiagonal matrix |

f07jec | Solves a real symmetric positive definite tridiagonal system using the modified Cholesky factorization computed by nag_dpttrf (f07jdc) |

f07jgc | Computes the reciprocal of the condition number of a real symmetric positive definite tridiagonal system using the modified Cholesky factorization computed by nag_dpttrf (f07jdc) |

f07jhc | Refined solution with error bounds of real symmetric positive definite tridiagonal system of linear equations, multiple right-hand sides |

f07jnc | Computes the solution to a complex Hermitian positive definite tridiagonal system of linear equations |

f07jpc | Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite tridiagonal system of linear equations |

f07jrc | Computes the modified Cholesky factorization of a complex Hermitian positive definite tridiagonal matrix |

f07jsc | Solves a complex Hermitian positive definite tridiagonal system using the modified Cholesky factorization computed by nag_zpttrf (f07jrc) |

f07juc | Computes the reciprocal of the condition number of a complex Hermitian positive definite tridiagonal system using the modified Cholesky factorization computed by nag_zpttrf (f07jrc) |

f07jvc | Refined solution with error bounds of complex Hermitian positive definite tridiagonal system of linear equations, multiple right-hand sides |

f07mac | Computes the solution to a real symmetric system of linear equations |

f07mbc | Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations |

f07mnc | Computes the solution to a complex Hermitian system of linear equations |

f07mpc | Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations |

f07nnc | Computes the solution to a complex symmetric system of linear equations |

f07npc | Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations |

f07pac | Computes the solution to a real symmetric system of linear equations, packed storage |

f07pbc | Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations, packed storage |

f07pnc | Computes the solution to a complex Hermitian system of linear equations, packed storage |

f07ppc | Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations, packed storage |

f07qnc | Computes the solution to a complex symmetric system of linear equations, packed storage |

f07qpc | Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations, packed storage |

f08aac | Solves an overdetermined or underdetermined real linear system |

f08anc | Solves an overdetermined or underdetermined complex linear system |

f08bac | Computes the minimum-norm solution to a real linear least squares problem |

f08bfc | $QR$ factorization of real general rectangular matrix with column pivoting, using BLAS-3 |

f08bhc | Reduces a real upper trapezoidal matrix to upper triangular form |

f08bkc | Apply orthogonal transformation determined by nag_dtzrzf (f08bhc) |

f08bnc | Computes the minimum-norm solution to a complex linear least squares problem |

f08btc | $QR$ factorization of complex general rectangular matrix with column pivoting, using BLAS-3 |

f08bvc | Reduces a complex upper trapezoidal matrix to upper triangular form |

f08bxc | Apply unitary transformation determined by nag_ztzrzf (f08bvc) |

f08cec | $QL$ factorization of real general rectangular matrix |

f08cfc | Form all or part of orthogonal $Q$ from $QL$ factorization determined by nag_dgeqlf (f08cec) |

f08cgc | Apply orthogonal transformation determined by nag_dgeqlf (f08cec) |

f08chc | $RQ$ factorization of real general rectangular matrix |

f08cjc | Form all or part of orthogonal $Q$ from $RQ$ factorization determined by nag_dgerqf (f08chc) |

f08ckc | Apply orthogonal transformation determined by nag_dgerqf (f08chc) |

f08csc | $QL$ factorization of complex general rectangular matrix |

f08ctc | Form all or part of orthogonal $Q$ from $QL$ factorization determined by nag_zgeqlf (f08csc) |

f08cuc | Apply unitary transformation determined by nag_zgeqlf (f08csc) |

f08cvc | $RQ$ factorization of complex general rectangular matrix |

f08cwc | Form all or part of orthogonal $Q$ from $RQ$ factorization determined by nag_zgerqf (f08cvc) |

f08cxc | Apply unitary transformation determined by nag_zgerqf (f08cvc) |

f08fac | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix |

f08fbc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix |

f08fdc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) |

f08flc | Computes 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 |

f08fnc | Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix |

f08fpc | Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix |

f08frc | Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) |

f08gac | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage |

f08gbc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage |

f08gnc | Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage |

f08gpc | Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage |

f08hac | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix |

f08hbc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix |

f08hnc | Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix |

f08hpc | Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix |

f08jac | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix |

f08jbc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix |

f08jdc | Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) |

f08jhc | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) |

f08jlc | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) |

f08jvc | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) |

f08jyc | Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) |

f08kac | Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition |

f08kbc | Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors |

f08kcc | Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition (divide-and-conquer) |

f08kdc | Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) |

f08khc | Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (preconditioned Jacobi) |

f08kjc | Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (fast Jacobi) |

f08knc | Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition |

f08kpc | Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors |

f08kqc | Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition (divide-and-conquer) |

f08krc | Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) |

f08mdc | Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) |

f08nac | Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix |

f08nbc | Computes 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 |

f08nnc | Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix |

f08npc | Computes 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 |

f08pac | Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors |

f08pbc | Computes 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 |

f08pnc | Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors |

f08ppc | Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues |

f08sac | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem |

f08sbc | Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem |

f08scc | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) |

f08snc | Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem |

f08spc | Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem |

f08sqc | Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) |

f08tac | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage |

f08tbc | Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage |

f08tcc | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) |

f08tnc | Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage |

f08tpc | Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage |

f08tqc | Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer) |

f08uac | Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem |

f08ubc | Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem |

f08ucc | Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) |

f08unc | Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem |

f08upc | Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem |

f08uqc | Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) |

f08vec | Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a real matrix pair |

f08vsc | Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a complex matrix pair |

f08wac | Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors |

f08wbc | Computes, 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 |

f08wnc | Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors |

f08wpc | Computes, 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 |

f08xac | Computes, 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 |

f08xbc | Computes, 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 |

f08xnc | Computes, 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 |

f08xpc | Computes, 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 |

f08yec | Computes the generalized singular value decomposition of a real upper triangular (or trapezoidal) matrix pair |

f08yfc | Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation |

f08ygc | Reorders 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 |

f08yhc | Solves the real-valued generalized Sylvester equation |

f08ylc | Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a real matrix pair in generalized real Schur canonical form |

f08ysc | Computes the generalized singular value decomposition of a complex upper triangular (or trapezoidal) matrix pair |

f08ytc | Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation |

f08yuc | Reorders 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 |

f08yvc | Solves the complex generalized Sylvester equation |

f08yyc | Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur canonical form |

f08zec | Computes a generalized $QR$ factorization of a real matrix pair |

f08zfc | Computes a generalized $RQ$ factorization of a real matrix pair |

f08zsc | Computes a generalized $QR$ factorization of a complex matrix pair |

f08ztc | Computes a generalized $RQ$ factorization of a complex matrix pair |

f11bdc | Real sparse nonsymmetric linear systems, setup for nag_sparse_nsym_basic_solver (f11bec) |

f11bec | Real sparse nonsymmetric linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method |

f11bfc | Real sparse nonsymmetric linear systems, diagnostic for nag_sparse_nsym_basic_solver (f11bec) |

f11brc | Complex sparse non-Hermitian linear systems, setup for nag_sparse_nherm_basic_solver (f11bsc) |

f11bsc | Complex sparse non-Hermitian linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method |

f11btc | Complex sparse non-Hermitian linear systems, diagnostic for nag_sparse_nherm_basic_solver (f11bsc) |

f11dbc | Solution of linear system involving incomplete $LU$ preconditioning matrix generated by nag_sparse_nsym_fac (f11dac) |

f11ddc | Solution of linear system involving preconditioning matrix generated by applying SSOR to real sparse nonsymmetric matrix |

f11dkc | Real sparse nonsymmetric linear systems, line Jacobi preconditioner |

f11dnc | Complex sparse non-Hermitian linear systems, incomplete $LU$ factorization |

f11dpc | Solution of complex linear system involving incomplete $LU$ preconditioning matrix generated by nag_sparse_nherm_fac (f11dnc) |

f11dqc | Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by nag_sparse_nherm_fac (f11dnc) (Black Box) |

f11drc | Solution of linear system involving preconditioning matrix generated by applying SSOR to complex sparse non-Hermitian matrix |

f11dsc | Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, Jacobi or SSOR preconditioner Black Box |

f11dxc | Complex sparse nonsymmetric linear systems, line Jacobi preconditioner |

f11gdc | Real sparse symmetric linear systems, setup for nag_sparse_sym_basic_solver (f11gec) |

f11gec | Real sparse symmetric linear systems, preconditioned conjugate gradient or Lanczos method or the MINRES algorithm |

f11gfc | Real sparse symmetric linear systems, diagnostic for nag_sparse_sym_basic_solver (f11gec) |

f11grc | Complex sparse Hermitian linear systems, setup for nag_sparse_herm_basic_solver (f11gsc) |

f11gsc | Complex sparse Hermitian linear systems, preconditioned conjugate gradient or Lanczos |

f11gtc | Complex sparse Hermitian linear systems, diagnostic for nag_sparse_herm_basic_solver (f11gsc) |

f11jbc | Solution of linear system involving incomplete Cholesky preconditioning matrix generated by nag_sparse_sym_chol_fac (f11jac) |

f11jdc | Solution of linear system involving preconditioning matrix generated by applying SSOR to real sparse symmetric matrix |

f11jnc | Complex sparse Hermitian matrix, incomplete Cholesky factorization |

f11jpc | Solution of complex linear system involving incomplete Cholesky preconditioning matrix generated by nag_sparse_herm_chol_fac (f11jnc) |

f11jqc | Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, preconditioner computed by nag_sparse_herm_chol_fac (f11jnc) (Black Box) |

f11jrc | Solution of linear system involving preconditioning matrix generated by applying SSOR to complex sparse Hermitian matrix |

f11jsc | Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) |

f11xac | Real sparse nonsymmetric matrix vector multiply |

f11xec | Real sparse symmetric matrix vector multiply |

f11xnc | Complex sparse non-Hermitian matrix vector multiply |

f11xsc | Complex sparse Hermitian matrix vector multiply |

f11znc | Complex sparse non-Hermitian matrix reorder function |

f11zpc | Complex sparse Hermitian matrix reorder function |

g01anc | Calculates approximate quantiles from a data stream of known size |

g01apc | Calculates approximate quantiles from a data stream of unknown size |

g01hcc | Computes probabilities for the bivariate Student's $t$-distribution |

g01kkc | Calculates a vector of values for the probability density function of the gamma distribution at chosen points |

g01kqc | Calculates a vector of values for the probability density function of the Normal distribution at chosen points |

g01sac | Computes a vector of probabilities for the standard Normal distribution |

g01sbc | Computes a vector of probabilities for Student's $t$-distribution |

g01scc | Computes a vector of probabilities for ${\chi}^{2}$ distribution |

g01sdc | Computes a vector of probabilities for $F$-distribution |

g01sec | Computes a vector of probabilities for the beta distribution |

g01sfc | Computes a vector of probabilities for the gamma distribution |

g01sjc | Computes a vector of the binomial distribution |

g01skc | Computes a vector of the Poisson distribution |

g01slc | Computes a vector of the hypergeometeric distribution |

g01tac | Computes a vector of deviates for the standard Normal distribution |

g01tbc | Computes a vector of deviates for Student's $t$-distribution |

g01tcc | Computes a vector of deviates for ${\chi}^{2}$ distribution |

g01tdc | Computes deviates for $F$-distribution |

g01tec | Computes a vector of deviates for the beta distribution |

g01tfc | Computes a vector of deviates for the gamma distribution |

g02abc | Computes the nearest correlation matrix to a real square matrix, augmented nag_nearest_correlation (g02aac) to incorporate weights and bounds |

g02aec | Computes the nearest correlation matrix with $k$-factor structure to a real square matrix |

g02qfc | Quantile linear regression, simple interface, independent, identically distributed (IID) errors |

g02qgc | Quantile linear regression, comprehensive interface |

g02zkc | Option setting function for nag_regsn_quant_linear (g02qgc) |

g02zlc | Option getting function for nag_regsn_quant_linear (g02qgc) |

g05kkc | Primes a pseudorandom number generator for generating multiple streams using skip-ahead, skipping ahead a power of $2$ |

g05nec | Pseudorandom sample, without replacement, unequal weights |

g07gac | Outlier detection using method of Peirce, raw data or single variance supplied |

g07gbc | Outlier detection using method of Peirce, two variances supplied |

g08chc | Calculates the Anderson–Darling goodness-of-fit test statistic |

g08cjc | Calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of uniformly distributed data |

g08ckc | Calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of a fully-unspecified Normal distribution |

g08clc | Calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of an unspecified exponential distribution |

g12abc | Computes rank statistics for comparing survival curves |

s14cbc | Logarithm of the beta function $\mathrm{ln}\left(B,a,b\right)$ |

s14ccc | Incomplete beta function ${I}_{x}\left(a,b\right)$ and its complement $1-{I}_{x}$ |

s17aqc | Bessel function vectorized ${Y}_{0}\left(x\right)$ |

s17arc | Bessel function vectorized ${Y}_{1}\left(x\right)$ |

s17asc | Bessel function vectorized ${J}_{0}\left(x\right)$ |

s17atc | Bessel function vectorized ${J}_{1}\left(x\right)$ |

s17auc | Airy function vectorized $\mathrm{Ai}\left(x\right)$ |

s17avc | Airy function vectorized $\mathrm{Bi}\left(x\right)$ |

s17awc | Airy function vectorized ${\mathrm{Ai}}^{\prime}\left(x\right)$ |

s17axc | Airy function vectorized ${\mathrm{Bi}}^{\prime}\left(x\right)$ |

s18aqc | Modified Bessel function vectorized ${K}_{0}\left(x\right)$ |

s18arc | Modified Bessel function vectorized ${K}_{1}\left(x\right)$ |

s18asc | Modified Bessel function vectorized ${I}_{0}\left(x\right)$ |

s18atc | Modified Bessel function vectorized ${I}_{1}\left(x\right)$ |

s18cqc | Scaled modified Bessel function vectorized ${e}^{x}{K}_{0}\left(x\right)$ |

s18crc | Scaled modified Bessel function vectorized ${e}^{x}{K}_{1}\left(x\right)$ |

s18csc | Scaled modified Bessel function vectorized ${e}^{-\left|x\right|}{I}_{0}\left(x\right)$ |

s18ctc | Scaled modified Bessel function vectorized ${e}^{-\left|x\right|}{I}_{1}\left(x\right)$ |

s19anc | Kelvin function vectorized $\mathrm{ber}x$ |

s19apc | Kelvin function vectorized $\mathrm{bei}x$ |

s19aqc | Kelvin function vectorized $\mathrm{ker}x$ |

s19arc | Kelvin function vectorized $\mathrm{kei}x$ |

s20aqc | Fresnel integral vectorized $S\left(x\right)$ |

s20arc | Fresnel integral vectorized $C\left(x\right)$ |

s30nbc | Heston's model option pricing formula with Greeks |

The following functions have been withdrawn from the NAG C Library at Mark 23. Warning of their withdrawal was included in the NAG C Library Manual at Mark 9, together with advice on which functions to use instead. See the document ‘Advice on Replacement Calls for Withdrawn/Superseded Functions’ for more detailed guidance.

The functions listed below are scheduled for withdrawal from the NAG C Library, because improved functions have now been included in the Library. You are advised to stop using functions which are scheduled for withdrawal and to use recommended replacement functions instead. See the document ‘Advice on Replacement Calls for Withdrawn/Superseded Functions’ for more detailed guidance, including advice on how to change a call to the old function into a call to its recommended replacement.

The following functions will be withdrawn at Mark 24.

The following functions have been superseded, but will not be withdrawn from the Library until Mark 25 at the earliest.