f11 Chapter Contents
f11 Chapter Introduction (PDF version)
NAG C Library Manual

NAG Library Chapter Introduction

f11 – Large Scale Linear Systems

+ Contents

1  Scope of the Chapter

This chapter provides functions for the solution of large sparse systems of simultaneous linear equations. These include iterative methods for real nonsymmetric and symmetric linear systems and direct methods for general real linear systems. Further direct methods are currently available in Chapters f01 and f04.

2  Background to the Problems

This section is only a brief introduction to the solution of sparse linear systems. For a more detailed discussion see for example Duff et al. (1986) and Demmel et al. (1999) for direct methods, or Barrett et al. (1994) for iterative methods.

2.1  Sparse Matrices and Their Storage

A matrix A may be described as sparse if the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations involving zero elements.
If A is sparse, and the chosen algorithm requires the matrix coefficients to be stored, a significant saving in storage can often be made by storing only the nonzero elements. A number of different formats may be used to represent sparse matrices economically. These differ according to the amount of storage required, the amount of indirect addressing required for fundamental operations such as matrix-vector products, and their suitability for vector and/or parallel architectures. For a survey of some of these storage formats see Barrett et al. (1994).
Black Box functions are those based on fixed storage formats. Three fixed storage formats for sparse matrices are currently used. These are known as coordinate storage (CS) format, symmetric coordinate storage (SCS) format and compressed column storage (CCS) format.

2.1.1  Co-ordinate storage (CS) format

This storage format represents a sparse matrix A, with nnz nonzero elements, in terms of three one-dimensional arrays – a double array a and two Integer arrays irow and icol. These arrays are all of dimension at least nnz. a contains the nonzero elements themselves, while irow and icol store the corresponding row and column indices respectively.
For example, the matrix
A= 1 -2 -1 -1 -3 0 -1 0 0 -4 3 0 0 0 2 2 0 4 1 1 -2 0 0 0 1
might be represented in the arrays a, irow and icol as
Notes
(i) The general format specifies no ordering of the array elements, but some functions may impose a specific ordering. For example, the nonzero elements may be required to be ordered by increasing row index and by increasing column index within each row, as in the example above. nag_sparse_nsym_sort (f11zac) is a utility function provided to order the elements appropriately (see Section 2.2).
(ii) With this storage format it is possible to enter duplicate elements. These may be interpreted in various ways (e.g., raising an error, ignoring all but the first entry, all but the last, or summing).

2.1.2  Symmetric coordinate storage (SCS) format

This storage format is suitable for symmetric and Hermitian matrices, and is identical to the CS format described in Section 2.1.1, except that only the lower triangular nonzero elements are stored. Thus, for example, the matrix
A= 4 -1 -0 -0 -1 2 1 5 0 2 0 0 0 0 2 1 0 -1 0 2 1 3 1 0 -1 0 0 1 4 0 2 0 -1 0 0 3
might be represented in the arrays a, irow and icol as

2.1.3  Compressed column storage (CCS) format

This storage format also uses three one-dimensional arrays – a double array a and two Integer arrays irowix and icolzp. The array a and irowix are of dimension at least nnz, while icolzp is of dimension at least n+1. a contains the nonzero elements, going down the first column, then the second and so on. For example, the matrix in Section 2.1.1 above will be represented by irowix records the row index for each entry in a, so the same matrix will have icolzp records the index into a which starts each new column. The last entry of icolzp is equal to nnz+1. An empty column (one filled with zeros, that is) is signalled by an index that is the same as the next non-empty column, or nnz+1 if all subsequent columns are empty. The above example corresponds to
The example in Section 2.1.2 above will be represented by

2.2  Direct Methods

Direct methods for the solution of the linear algebraic system
Ax=b (1)
aim to determine the solution vector x in a fixed number of arithmetic operations, which is determined a priori by the number of unknowns. For example, an LU factorization of A followed by forward and backward substitution is a direct method for (1).
If the matrix A is sparse it is possible to design direct methods which exploit the sparsity pattern and are therefore much more computationally efficient than the algorithms in Chapter f07, which in general take no account of sparsity. However, if the matrix is very large and sparse, then iterative methods, with an appropriate preconditioner, (see Section 2.3) may be more efficient still.
This chapter provides a direct LU factorization method for sparse real systems. This method is based on special coding for supernodes, broadly defined as groups of consecutive columns with the same nonzero structure, which enables use of dense BLAS kernels. The algorithms contained here come from the SuperLU software suite (see Demmel et al. (1999)). An important requirement of sparse LU factorization is keeping the factors as sparse as possible. It is well known that certain column orderings can produce much sparser factorizations than the normal left-to-right ordering. It is well worth the effort, then, to find such column orderings since they reduce both storage requirements of the factors, the time taken to compute them and the time taken to solve the linear system. The row reorderings, demanded by partial pivoting in order to keep the factorization stable, can further complicate the choice of the column ordering, but quite good and fast algorithms have been developed to make possible a fairly reliable computation of an appropriate column ordering for any sparsity pattern. We provide one such algorithm (known in the literature as COLAMD) through one function in the suite. Similar to the case for dense matrices, functions are provided to compute the LU factorization with partial row pivoting for numerical stability, solve (1) by performing the forward and backward substitutions for multiple right hand side vectors, refine the solution, minimize the backward error and estimate the forward error of the solutions, compute norms, estimate condition numbers and perform diagnostics of the factorization. For more details see Section 3.4.
Further direct methods may be found in Chapters f01, f04 and f07.

2.3  Iterative Methods

In contrast to the direct methods discussed in Section 2.2, iterative methods for (1) approach the solution through a sequence of approximations until some user-specified termination criterion is met or until some predefined maximum number of iterations has been reached. The number of iterations required for convergence is not generally known in advance, as it depends on the accuracy required, and on the matrix A – its sparsity pattern, conditioning and eigenvalue spectrum.
Faster convergence can often be achieved using a preconditioner (see Golub and Van Loan (1996) and Barrett et al. (1994)). A preconditioner maps the original system of equations onto a different system
A-x-=b-, (2)
which hopefully exhibits better convergence characteristics. For example, the condition number of the matrix A- may be better than that of A, or it may have eigenvalues of greater multiplicity.
An unsuitable preconditioner or no preconditioning at all may result in a very slow rate or lack of convergence. However, preconditioning involves a trade-off between the reduction in the number of iterations required for convergence and the additional computational costs per iteration. Setting up a preconditioner may also involve non-negligible overheads. The application of preconditioners to real nonsymmetric and real symmetric systems of equations is further considered in Sections 2.4 and 2.5.

2.4  Iterative Methods for Real Nonsymmetric Linear Systems

Many of the most effective iterative methods for the solution of (1) lie in the class of non-stationary Krylov subspace methods (see Barrett et al. (1994)). For real nonsymmetric matrices this class includes:
Here we just give a brief overview of these algorithms as implemented in this chapter.
RGMRES is based on the Arnoldi method, which explicitly generates an orthogonal basis for the Krylov subspace spanAkr0, k=0,1,2,, where r0 is the initial residual. The solution is then expanded onto the orthogonal basis so as to minimize the residual norm. For real nonsymmetric matrices the generation of the basis requires a ‘long’ recurrence relation, resulting in prohibitive computational and storage costs. RGMRES limits these costs by restarting the Arnoldi process from the latest available residual every m iterations. The value of m is chosen in advance and is fixed throughout the computation. Unfortunately, an optimum value of m cannot easily be predicted.
CGS is a development of the bi-conjugate gradient method where the nonsymmetric Lanczos method is applied to reduce the coefficient matrix to tridiagonal form: two bi-orthogonal sequences of vectors are generated starting from the initial residual r0 and from the shadow residual r^0 corresponding to the arbitrary problem AHx^=b^, where b^ is chosen so that r0=r^0. In the course of the iteration, the residual and shadow residual ri=PiAr0 and r^i=PiAHr^0 are generated, where Pi is a polynomial of order i, and bi-orthogonality is exploited by computing the vector product ρ i = r ^ i , r i = Pi AH r ^ 0 P i A r 0 = r ^ 0 , P i 2 A r 0 . Applying the ‘contraction’ operator PiA twice, the iteration coefficients can still be recovered without advancing the solution of the shadow problem, which is of no interest. The CGS method often provides fast convergence; however, there is no reason why the contraction operator should also reduce the once reduced vector PiAr0: this can lead to a highly irregular convergence.
Bi-CGSTAB is similar to the CGS method. However, instead of generating the sequence Pi2Ar0, it generates the sequence QiAPiAr0 where the QiA are polynomials chosen to minimize the residual after the application of the contraction operator PiA. Two main steps can be identified for each iteration: an OR (Orthogonal Residuals) step where a basis of order  is generated by a Bi-CG iteration and an MR (Minimum Residuals) step where the residual is minimized over the basis generated, by a method similar to GMRES. For =1, the method corresponds to the Bi-CGSTAB method of Van der Vorst (1989). However, as  increases, numerical instabilities may arise.
The transpose-free quasi-minimal residual method (TFQMR) (see Freund and Nachtigal (1991) and Freund (1993)) is conceptually derived from the CGS method. The residual is minimized over the space of the residual vectors generated by the CGS iterations under the simplifying assumption that residuals are almost orthogonal. In practice, this is not the case but theoretical analysis has proved the validity of the method. This has the effect of remedying the rather irregular convergence behaviour with wild oscillations in the residual norm that can degrade the numerical performance and robustness of the CGS method. In general, the TFQMR method can be expected to converge at least as fast as the CGS method, in terms of number of iterations, although each iteration involves a higher operation count. When the CGS method exhibits irregular convergence, the TFQMR method can produce much smoother, almost monotonic convergence curves. However, the close relationship between the CGS and TFQMR method implies that the overall speed of convergence is similar for both methods. In some cases, the TFQMR method may converge faster than the CGS method.
Faster convergence can usually be achieved by using a preconditioner. A left preconditioner M-1 can be used by the RGMRES, CGS and TFQMR methods, such that A-=M-1AIn in (2), where In is the identity matrix of order n; a right preconditioner M-1 can be used by the Bi-CGSTAB method, such that A-=AM-1In. These are formal definitions, used only in the design of the algorithms; in practice, only the means to compute the matrix-vector products v=Au and v=AHu (the latter only being required when an estimate of A1 or A is computed internally), and to solve the preconditioning equations Mv=u are required, that is, explicit information about M, or its inverse is not required at any stage.
Preconditioning matrices M are typically based on incomplete factorizations (see Meijerink and Van der Vorst (1981)), or on the approximate inverses occurring in stationary iterative methods (see Young (1971)). A common example is the incomplete LU factorization
M=PLDUQ=A-R
where L is lower triangular with unit diagonal elements, D is diagonal, U is upper triangular with unit diagonals, P and Q are permutation matrices, and R is a remainder matrix. A zero-fill incomplete LU factorization is one for which the matrix
S=PL+D+UQ
has the same pattern of nonzero entries as A. This is obtained by discarding any fill elements (nonzero elements of S arising during the factorization in locations where A has zero elements). Allowing some of these fill elements to be kept rather than discarded generally increases the accuracy of the factorization at the expense of some loss of sparsity. For further details see Barrett et al. (1994).

2.5  Iterative Methods for Real Symmetric Linear Systems

Three of the best known iterative methods applicable to real symmetric linear systems are the conjugate gradient (CG) method (see Hestenes and Stiefel (1952) and Golub and Van Loan (1996)) and Lanczos type methods based on SYMMLQ and MINRES (see Paige and Saunders (1975)).
For the CG method the matrix A should ideally be positive definite. The application of CG to indefinite matrices may lead to failure, or to lack of convergence. The SYMMLQ and MINRES methods are suitable for both positive definite and indefinite symmetric matrices. They are more robust than CG, but less efficient when A is positive definite.
The methods start from the residual r0=b-Ax0, where x0 is an initial estimate for the solution (often x0=0), and generate an orthogonal basis for the Krylov subspace spanAkr0, for k=0,1,, by means of three-term recurrence relations (see Golub and Van Loan (1996)). A sequence of symmetric tridiagonal matrices Tk is also generated. Here and in the following, the index k denotes the iteration count. The resulting symmetric tridiagonal systems of equations are usually more easily solved than the original problem. A sequence of solution iterates xk is thus generated such that the sequence of the norms of the residuals rk converges to a required tolerance. Note that, in general, the convergence is not monotonic.
In exact arithmetic, after n iterations, this process is equivalent to an orthogonal reduction of A to symmetric tridiagonal form, Tn=QTAQ; the solution xn would thus achieve exact convergence. In finite-precision arithmetic, cancellation and round-off errors accumulate causing loss of orthogonality. These methods must therefore be viewed as genuinely iterative methods, able to converge to a solution within a prescribed tolerance.
The orthogonal basis is not formed explicitly in either method. The basic difference between the methods lies in the method of solution of the resulting symmetric tridiagonal systems of equations: the CG method is equivalent to carrying out an LDLT (Cholesky) factorization whereas the Lanczos method (SYMMLQ) uses an LQ factorization. The MINRES method on the other hand minimizes the residual into 2-norm.
A preconditioner for these methods must be symmetric and positive definite, i.e., representable by M=EET, where M is nonsingular, and such that A-=E-1AE-TIn in (2), where In is the identity matrix of order n. These are formal definitions, used only in the design of the algorithms; in practice, only the means to compute the matrix-vector products v=Au and to solve the preconditioning equations Mv=u are required.
Preconditioning matrices M are typically based on incomplete factorizations (see Meijerink and Van der Vorst (1977)), or on the approximate inverses occurring in stationary iterative methods (see Young (1971)). A common example is the incomplete Cholesky factorization
M=PLDLTPT=A-R
where P is a permutation matrix, L is lower triangular with unit diagonal elements, D is diagonal and R is a remainder matrix. A zero-fill incomplete Cholesky factorization is one for which the matrix
S=PL+D+LTPT
has the same pattern of nonzero entries as A. This is obtained by discarding any fill elements (nonzero elements of S arising during the factorization in locations where A has zero elements). Allowing some of these fill elements to be kept rather than discarded generally increases the accuracy of the factorization at the expense of some loss of sparsity. For further details see Barrett et al. (1994).

3  Recommendations on Choice and Use of Available Functions

3.1  Types of Function Available

The direct method functions available in this chapter largely follow the LAPACK scheme in that four different functions separately handle the tasks of factorizing, solving, refining and condition number estimating. See Section 3.4.
The iterative method functions available in this chapter divide essentially into two types: utility functions and Black Box functions.
At present there are suites of basic functions for real symmetric and nonsymmetric systems, and for complex non-Hermitian systems.
Utility functions perform such tasks as initializing the preconditioning matrix M or computing matrix-vector products, for particular preconditioners and matrix storage formats.
Black Box functions provide easy-to-use functions for particular preconditioners and sparse matrix storage formats.

3.2  Iterative Methods for Real Nonsymmetric Linear Systems

In general, it is not possible to recommend one of these methods (RGMRES, CGS, Bi-CGSTAB, or TFQMR) in preference to another. RGMRES is popular, but requires the most storage, and can easily stagnate when the size m of the orthogonal basis is too small, or the preconditioner is not good enough. CGS can be the fastest method, but the computed residuals can exhibit instability which may greatly affect the convergence and quality of the solution. Bi-CGSTAB seems robust and reliable, but it can be slower than the other methods. TFQMR can be viewed as a more robust variant of the CGS method: it shares the CGS method speed but avoids the CGS fluctuations in the residual, which may give, rise to instability. Some further discussion of the relative merits of these methods can be found in Barrett et al. (1994).
The utility functions provided for real nonsymmetric matrices use the coordinate storage (CS) format described in Section 2.1.1. nag_sparse_nsym_fac (f11dac) computes a preconditioning matrix based on incomplete LU factorization. The amount of fill-in occurring in the incomplete factorization can be controlled by specifying either the level of fill, or the drop tolerance. Partial or complete pivoting may optionally be employed, and the factorization can be modified to preserve row-sums.
nag_sparse_nsym_sort (f11zac) orders the nonzero elements of a real sparse nonsymmetric matrix stored in general CS format.
The Black Box function nag_sparse_nsym_fac_sol (f11dcc) solves a real sparse nonsymmetric linear system, represented in CS format, using RGMRES, CGS, Bi-CGSTAB, or TFQMR, with incomplete LU preconditioning. nag_sparse_nsym_sol (f11dec) is similar, but has options for no preconditioning, Jacobi preconditioning or SSOR preconditioning.

3.3  Iterative Methods for Real Symmetric Linear Systems

The utility functions provided for real symmetric matrices use the symmetric coordinate storage (SCS) format described in Section 2.1.2. nag_sparse_sym_chol_fac (f11jac) computes a preconditioning matrix based on incomplete Cholesky factorization. The amount of fill-in occurring in the incomplete factorization can be controlled by specifying either the level of fill, or the drop tolerance. Diagonal Markowitz pivoting may optionally be employed, and the factorization can be modified to preserve row-sums.
nag_sparse_sym_sort (f11zbc) orders the nonzero elements of a real sparse symmetric matrix stored in general SCS format.
The Black Box function nag_sparse_sym_chol_sol (f11jcc) solves a real sparse symmetric linear system, represented in SCS format, using a conjugate gradient or Lanczos method, with incomplete Cholesky preconditioning. nag_sparse_sym_sol (f11jec) is similar, but has options for no preconditioning, Jacobi preconditioning or SSOR preconditioning.

3.4  Direct Methods

The suite of functions nag_superlu_column_permutation (f11mdc), nag_superlu_lu_factorize (f11mec), nag_superlu_solve_lu (f11mfc), nag_superlu_condition_number_lu (f11mgc), nag_superlu_refine_lu (f11mhc), nag_superlu_matrix_product (f11mkc), nag_superlu_matrix_norm (f11mlc) and nag_superlu_diagnostic_lu (f11mmc) implement the COLAMD/SuperLU direct real sparse solver and associated utilities. You are expected to first call nag_superlu_column_permutation (f11mdc) to compute a suitable column permutation for the subsequent factorization by nag_superlu_lu_factorize (f11mec). nag_superlu_solve_lu (f11mfc) then solves the system of equations. A solution can be further refined by nag_superlu_refine_lu (f11mhc), which also minimizes the backward error and estimates a bound for the forward error in the solution. Diagnostics are provided by nag_superlu_condition_number_lu (f11mgc) which computes an estimate of the condition number of the matrix using the factorization output by nag_superlu_lu_factorize (f11mec), and nag_superlu_diagnostic_lu (f11mmc) which computes the reciprocal pivot growth (a numerical stability measure) of the factorization. The two utility functions, nag_superlu_matrix_product (f11mkc), which computes matrix-matrix products in the particular storage scheme demanded by the suite, and nag_superlu_matrix_norm (f11mlc) which computes quantities relating to norms of a matrix in that particular storage scheme, complete the suite.
Some other functions specifically designed for direct solution of sparse linear systems can currently be found in Chapters f01, f04 and f07. In particular, the following functions allow the direct solution of symmetric positive definite systems:
Variable band (skyline) nag_real_cholesky_skyline (f01mcc) and nag_real_cholesky_skyline_solve (f04mcc)
Functions for the solution of band and tridiagonal systems can be found in Chapters f04 and f07.

4  Decision Tree

Tree 1: Solvers

Do you have a real system and want to use a direct method? _
yes
f11mdc, f11mec and f11mfc
no
|
Symmetric positive definite? _
yes
Incomplete Cholesky preconditioner? _
yes
f11jac and f11jcc
| no
|
| f11jec
no
|
Incomplete LU preconditioner? _
yes
f11dac and f11dcc
no
|
f11dec

5  Functionality Index

Apply iterative refinement to the solution and compute error estimates, after factorizing the matrix of coefficients, 
    real sparse nonsymmetric matrix in CCS format nag_superlu_refine_lu (f11mhc)
Basic functions for complex Hermitian linear systems, 
    diagnostic function nag_sparse_herm_basic_diagnostic (f11gtc)
    setup function nag_sparse_herm_basic_setup (f11grc)
Basic functions for complex non-Hermitian linear systems, 
    diagnostic function nag_sparse_nherm_basic_diagnostic (f11btc)
    reverse communication RGMRES, CGS, Bi-CGSTAB() or TFQMR solver function nag_sparse_nherm_basic_solver (f11bsc)
    setup function nag_sparse_nherm_basic_setup (f11brc)
Basic functions for real nonsymmetric linear systems, 
    diagnostic function nag_sparse_nsym_basic_diagnostic (f11bfc)
    reverse communication RGMRES, CGS, Bi-CGSTAB() or TFQMR solver function nag_sparse_nsym_basic_solver (f11bec)
    setup function nag_sparse_nsym_basic_setup (f11bdc)
Basic functions for real symmetric linear systems, 
    diagnostic function nag_sparse_sym_basic_diagnostic (f11gfc)
    reverse communication CG or SYMMLQ solver nag_sparse_sym_basic_solver (f11gec)
    setup function nag_sparse_sym_basic_setup (f11gdc)
Basic routines for real sparse nonsymmetric linear systems 
    Matrix-matrix multiplier for real sparse nonsymmetric matrices in CCS format nag_superlu_matrix_product (f11mkc)
Black Box functions for complex Hermitian linear systems, 
    CG or SYMMLQ solver 
        with incomplete Cholesky preconditioning nag_sparse_herm_chol_sol (f11jqc)
        with no preconditioning, Jacobi or SSOR preconditioning nag_sparse_herm_sol (f11jsc)
Black Box functions for complex non-Hermitian linear systems, 
    RGMRES, CGS, Bi-CGSTAB() or TFQMR solver 
        with incomplete LU preconditioning nag_sparse_nherm_fac_sol (f11dqc)
        with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_nherm_sol (f11dsc)
Black Box functions for real nonsymmetric linear systems, 
    RGMRES, CGS, Bi-CGSTAB() or TFQMR solver 
        with incomplete LU preconditioning nag_sparse_nsym_fac_sol (f11dcc)
        with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_nsym_sol (f11dec)
Black Box functions for real symmetric linear systems, 
    CG or SYMMLQ solver 
        with incomplete Cholesky preconditioning nag_sparse_sym_chol_sol (f11jcc)
        with no preconditioning, Jacobi, or SSOR preconditioning nag_sparse_sym_sol (f11jec)
Compute a norm or the element of largest absolute value, 
    real sparse nonsymmetric matrix in CCS format nag_superlu_matrix_norm (f11mlc)
Condition number estimation, after factorizing the matrix of coefficients, 
    real sparse nonsymmetric matrix in CCS format nag_superlu_condition_number_lu (f11mgc)
LU factorization, 
    diagnostic routine, 
        real sparse nonsymmetric matrix in CCS format nag_superlu_diagnostic_lu (f11mmc)
    real sparse nonsymmetric matrix in CCS format nag_superlu_lu_factorize (f11mec)
    setup routine, 
        real sparse nonsymmetric matrices in CCS format nag_superlu_column_permutation (f11mdc)
matrix-vector multiplier for complex Hermitian matrices in SCS format nag_sparse_herm_matvec (f11xsc)
reverse communication CG or SYMMLQ solver function nag_sparse_herm_basic_solver (f11gsc)
Solution of simultaneous linear equations, after factorizing the matrix of coefficients, 
    real sparse nonsymmetric matrix in CCS format nag_superlu_solve_lu (f11mfc)
Utility function for complex Hermitian linear systems, 
    incomplete Cholesky factorization nag_sparse_herm_chol_fac (f11jnc)
    solver for linear systems involving preconditioning matrix from nag_sparse_herm_chol_fac (f11jnc) nag_sparse_herm_precon_ichol_solve (f11jpc)
    solver for linear systems involving SSOR preconditioning matrix nag_sparse_herm_precon_ssor_solve (f11jrc)
    sort function for complex Hermitian matrices in SCS format nag_sparse_herm_sort (f11zpc)
Utility function for complex non-Hermitian linear systems, 
    incomplete LU factorization nag_sparse_nherm_fac (f11dnc)
    matrix-vector multiplier for complex non-Hermitian matrices in CS format nag_sparse_nherm_matvec (f11xnc)
    solver for linear systems involving iterated Jacobi method nag_sparse_nherm_jacobi (f11dxc)
    solver for linear systems involving preconditioning matrix from nag_sparse_nherm_fac (f11dnc) nag_sparse_nherm_precon_ilu_solve (f11dpc)
    solver for linear systems involving SSOR preconditioning matrix nag_sparse_nherm_precon_ssor_solve (f11drc)
    sort function for complex non-Hermitian matrices in CS format nag_sparse_nherm_sort (f11znc)
Utility function for real nonsymmetric linear systems, 
    incomplete LU factorization nag_sparse_nsym_fac (f11dac)
    matrix-vector multiplier for real nonsymmetric matrices in CS format nag_sparse_nsym_matvec (f11xac)
    solver for linear systems involving iterated Jacobi method nag_sparse_nsym_jacobi (f11dkc)
    solver for linear systems involving preconditioning matrix from nag_sparse_nsym_fac (f11dac) nag_sparse_nsym_precon_ilu_solve (f11dbc)
    solver for linear systems involving SSOR preconditioning matrix nag_sparse_nsym_precon_ssor_solve (f11ddc)
    sort function for real nonsymmetric matrices in CS format nag_sparse_nsym_sort (f11zac)
Utility function for real symmetric linear systems, 
    incomplete Cholesky factorization nag_sparse_sym_chol_fac (f11jac)
    matrix-vector multiplier for real symmetric matrices in SCS format nag_sparse_sym_matvec (f11xec)
    solver for linear systems involving preconditioning matrix from nag_sparse_sym_chol_fac (f11jac) nag_sparse_sym_precon_ichol_solve (f11jbc)
    solver for linear systems involving SSOR preconditioning matrix nag_sparse_sym_precon_ssor_solve (f11jdc)
    sort function for real symmetric matrices in SCS format nag_sparse_sym_sort (f11zbc)

6  Functions Withdrawn or Scheduled for Withdrawal

None.

7  References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Demmel J W, Eisenstat S C, Gilbert J R, Li X S and Li J W H (1999) A supernodal approach to sparse partial pivoting SIAM J. Matrix Anal. Appl. 20 720–755
Duff I S, Erisman A M and Reid J K (1986) Direct Methods for Sparse Matrices Oxford University Press, London
Freund R W (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems Numer. Math. 60 315–339
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hestenes M and Stiefel E (1952) Methods of conjugate gradients for solving linear systems J. Res. Nat. Bur. Stand. 49 409–436
Meijerink J and Van der Vorst H (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math. Comput. 31 148–162
Meijerink J and Van der Vorst H (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems J. Comput. Phys. 44 134–155
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869
Sleijpen G L G and Fokkema D R (1993) BiCGSTAB for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

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NAG C Library Manual

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