f11bd:: Large Scale Linear Systems (NAG Toolbox)

nag_sparse_real_gen_basic_setup (f11bd) is a setup function which must be called before nag_sparse_real_gen_basic_solver (f11be), the iterative solver. The third function in the suite, nag_sparse_real_gen_basic_diag (f11bf), can be used to return additional information about the computation. A choice of methods is available:

restarted generalized minimum residual method (RGMRES);
conjugate gradient squared method (CGS);
bi-conjugate gradient stabilized ( $ℓ$ ) method (Bi-CGSTAB( $ℓ$ ));
transpose-free quasi-minimal residual method (TFQMR).

Restarted Generalized Minimum Residual Method (RGMRES)

The restarted generalized minimum residual method (RGMRES) (see Saad and Schultz (1986), Barrett et al. (1994) and Dias da Cunha and Hopkins (1994)) starts from the residual

r_{0} = b - A x_{0}

, where

x_{0}

is an initial estimate for the solution (often

x_{0} = 0

). An orthogonal basis for the Krylov subspace

span \{A^{k} r_{0}\}

, for

k = 0, 1, \dots

, is generated explicitly: this is referred to as Arnoldi's method (see Arnoldi (1951)). The solution is then expanded onto the orthogonal basis so as to minimize the residual norm

{‖b - A x‖}_{2}

. The lack of symmetry of

A

implies that the orthogonal basis is generated by applying a ‘long’ recurrence relation, whose length increases linearly with the iteration count. For all but the most trivial problems, computational and storage costs can quickly become prohibitive as the iteration count increases. RGMRES limits these costs by employing a restart strategy: every

m

iterations at most, the Arnoldi process is restarted from

r_{l} = b - A x_{l}

, where the subscript

l

denotes the last available iterate. Each group of

m

iterations is referred to as a ‘super-iteration’. The value of

m

is chosen in advance and is fixed throughout the computation. Unfortunately, an optimum value of

m

cannot easily be predicted.

Conjugate Gradient Squared Method (CGS)

The conjugate gradient squared method (CGS) (see Sonneveld (1989), Barrett et al. (1994) and Dias da Cunha and Hopkins (1994)) is a development of the bi-conjugate gradient method where the nonsymmetric Lanczos method is applied to reduce the coefficients matrix to real tridiagonal form: two bi-orthogonal sequences of vectors are generated starting from the residual

r_{0} = b - A x_{0}

, where

x_{0}

is an initial estimate for the solution (often

x_{0} = 0

) and from the shadow residual

{\hat{r}}_{0}

corresponding to the arbitrary problem

A^{T} \hat{x} = \hat{b}

, where

\hat{b}

can be any vector, but in practice is chosen so that

r_{0} = {\hat{r}}_{0}

. In the course of the iteration, the residual and shadow residual

r_{i} = P_{i} (A) r_{0}

and

{\hat{r}}_{i} = P_{i} (A^{T}) {\hat{r}}_{0}

are generated, where

P_{i}

is a polynomial of order

i

, and bi-orthogonality is exploited by computing the vector product

ρ_{i} = ({\hat{r}}_{i}, r_{i}) = (P_{i} (A^{T}) {\hat{r}}_{0}, P_{i} (A) r_{0}) = ({\hat{r}}_{0}, P_{i}^{2} (A) r_{0})

. Applying the ‘contraction’ operator

P_{i} (A)

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

P_{i} (A) r_{0}

: this may well lead to a highly irregular convergence which may result in large cancellation errors.

Bi-Conjugate Gradient Stabilized (ℓ) Method (Bi-CGSTAB(ℓ))

The bi-conjugate gradient stabilized (

ℓ

) method (Bi-CGSTAB(

ℓ

)) (see Van der Vorst (1989), Sleijpen and Fokkema (1993) and Dias da Cunha and Hopkins (1994)) is similar to the CGS method above. However, instead of generating the sequence

\{P_{i}^{2} (A) r_{0}\}

, it generates the sequence

\{Q_{i} (A) P_{i} (A) r_{0}\}

, where the

Q_{i} (A)

are polynomials chosen to minimize the residual after the application of the contraction operator

P_{i} (A)

. Two main steps can be identified for each iteration: an OR (Orthogonal Residuals) step where a basis of order

ℓ

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 akin to GMRES. For

ℓ = 1

, the method corresponds to the Bi-CGSTAB method of Van der Vorst (1989). For

ℓ > 1

, more information about complex eigenvalues of the iteration matrix can be taken into account, and this may lead to improved convergence and robustness. However, as

ℓ

increases, numerical instabilities may arise. For this reason, a maximum value of

ℓ = 10

is imposed, but probably

ℓ = 4

is sufficient in most cases.

Transpose-free Quasi-minimal Residual Method (TFQMR)

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.

General Considerations

Faster convergence can 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, say

\bar{A} \bar{x} = \bar{b},

(1)

with, hopefully, better characteristics with respect to its speed of convergence: for example, the condition number of the coefficients matrix can be improved or eigenvalues in its spectrum can be made to coalesce. An orthogonal basis for the Krylov subspace

span \{{\bar{A}}^{k} {\bar{r}}_{0}\}

, for

k = 0, 1, \dots

, is generated and the solution proceeds as outlined above. The algorithms used are such that the solution and residual iterates of the original system are produced, not their preconditioned counterparts. Note that 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. Also, setting up a preconditioner may involve non-negligible overheads.

A left preconditioner

M^{- 1}

can be used by the RGMRES, CGS and TFQMR methods, such that

\bar{A} = M^{- 1} A \sim I_{n}

in (1), where

I_{n}

is the identity matrix of order

n

; a right preconditioner

M^{- 1}

can be used by the Bi-CGSTAB(

ℓ

) method, such that

\bar{A} = A M^{- 1} \sim I_{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 = A u

and

v = A^{T} u

(the latter only being required when an estimate of

{‖A‖}_{1}

{‖A‖}_{\infty}

is computed internally), and to solve the preconditioning equations

M v = u

are required, i.e., explicit information about

M

, or its inverse is not required at any stage.

The first termination criterion

{‖r_{k}‖}_{p} \leq τ ({‖b‖}_{p} + {‖A‖}_{p} \times {‖x_{k}‖}_{p})

(2)

is available for all four methods. In (2),

p = 1

\infty ​ or ​ 2

and

τ

denotes a user-specified tolerance subject to

\max (10, \sqrt{n})

ε \leq τ < 1

, where

ε

is the machine precision. Facilities are provided for the estimation of the norm of the coefficients matrix

{‖A‖}_{1}

{‖A‖}_{\infty}

, when this is not known in advance, by applying Higham's method (see Higham (1988)). Note that

{‖A‖}_{2}

cannot be estimated internally. This criterion uses an error bound derived from backward error analysis to ensure that the computed solution is the exact solution of a problem as close to the original as the termination tolerance requires. Termination criteria employing bounds derived from forward error analysis are not used because any such criteria would require information about the condition number

κ (A)

which is not easily obtainable.

The second termination criterion

{‖{\bar{r}}_{k}‖}_{2} \leq τ ({‖{\bar{r}}_{0}‖}_{2} + σ_{1} (\bar{A}) \times {‖Δ {\bar{x}}_{k}‖}_{2})

(3)

is available for all methods except TFQMR. In (3),

σ_{1} (\bar{A}) = {‖\bar{A}‖}_{2}

is the largest singular value of the (preconditioned) iteration matrix

\bar{A}

. This termination criterion monitors the progress of the solution of the preconditioned system of equations and is less expensive to apply than criterion (2) for the Bi-CGSTAB(

ℓ

) method with

ℓ > 1

. Only the RGMRES method provides facilities to estimate

σ_{1} (\bar{A})

internally, when this is not supplied (see Further Comments).

Termination criterion (2) is the recommended choice, despite its additional costs per iteration when using the Bi-CGSTAB(

ℓ

) method with

ℓ > 1

. Also, if the norm of the initial estimate is much larger than the norm of the solution, that is, if

‖x_{0}‖ ≫ ‖x‖

, a dramatic loss of significant digits could result in complete lack of convergence. The use of criterion (2) will enable the detection of such a situation, and the iteration will be restarted at a suitable point. No such restart facilities are provided for criterion (3).

In general, it is not possible to recommend one method in preference to another. RGMRES is often used in the solution of systems arising from PDEs. On the other hand, it 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(

ℓ

) seems robust and reliable, but it can be slower than the other methods: if a preconditioner is used and

ℓ > 1

, Bi-CGSTAB(

ℓ

) computes the solution of the preconditioned system

{\bar{x}}_{k} = M x_{k}

: the preconditioning equations must be solved to obtain the required solution. The algorithm employed limits to

10 %

or less, when no intermediate monitoring is requested, the number of times the preconditioner has to be thus applied compared with the total number of applications of the preconditioner. 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. Also, when the termination criterion (2) is used, the CGS, Bi-CGSTAB(

ℓ

) and TFQMR methods will restart the iteration automatically when necessary in order to solve the given problem.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Loss of orthogonality in the RGMRES method, or of bi-orthogonality in the Bi-CGSTAB(

ℓ

) method may degrade the solution and speed of convergence. For both methods, the algorithms employed include checks on the basis vectors so that the number of basis vectors used for a given super-iteration may be less than the value specified in the input argument m. Also, if termination criterion (2) is used the CGS, Bi-CGSTAB(

ℓ

) and TFQMR methods will restart automatically the computation from the last available iterates, when the stability of the solution process requires it.

Termination criterion (3), when available, involves only the residual (or norm of the residual) produced directly by the iteration process: this may differ from the norm of the true residual

{\tilde{r}}_{k} = b - A x_{k}

, particularly when the norm of the residual is very small. Also, if the norm of the initial estimate of the solution is much larger than the norm of the exact solution, convergence can be achieved despite very large errors in the solution. On the other hand, termination criterion (3) is cheaper to use and inspects the progress of the actual iteration. Termination criterion (2) should be preferred in most cases, despite its slightly larger costs.

Example

function f11bd_example


fprintf('f11bd example results\n\n');

% Solve sparse system Ax = b iteratively using ILU preconditioner

% Define sparse matrix A and RHS B
n    = int64(8);
nz   = int64(24);
a    = zeros(3*nz, 1);
irow = zeros(3*nz, 1, 'int64');
icol = irow;
irow(1:nz) = int64(...
             [1; 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 4;
              5; 5; 5; 6; 6; 6; 7; 7; 7; 8; 8; 8]);
icol(1:nz) = int64(...
             [1; 4; 8; 1; 2; 5; 3; 6; 1; 3; 4; 7;
              2; 5; 7; 1; 3; 6; 3; 5; 7; 2; 6; 8]);
a(1:nz)    = [2;-1; 1; 4;-3; 2;-7; 2; 3;-4; 5; 5;
             -1; 8;-3;-6; 5; 2;-5;-1; 6;-1; 2; 3];
b          = [6; 8;-9;46;17;21;22;34];

% ILU preconditioner
%     Input parameters
lfill = int64(0);
dtol  = 0;
milu  = 'No modification';
ipivp = zeros(n, 1, 'int64');
ipivq = zeros(n, 1, 'int64');
% Compute preconditioner
[a, irow, icol, ipivp, ipivq, istr, idiag, nnzc, npivm, ifail] = ...
    f11da(...
          nz, a, irow, icol, lfill, dtol, milu, ipivp, ipivq);

% Iterative Solution Setup
%      Input parameters
method = 'BICGSTAB';
precon = 'Preconditioned';
lpoly  = int64(1);
tol    = sqrt(x02aj);
maxitn = int64(20);
anorm  = 0;
sigmax = 0;
monit  = int64(1);
lwork  = int64(6000);
% Setup iterative method
[lwreq, work, ifail] = ...
  f11bd(...
        method, precon, n, lpoly, tol, maxitn, anorm, sigmax, ...
        monit, lwork, 'norm_p', '1');

% Reverse communication loop calling iterative solver
irevcm = int64(0);
u      = zeros(8, 1);
v      = b;
wgt    = zeros(8, 1);

while (irevcm ~= 4)

  [irevcm, u, v, work, ifail] = f11be(...
                                      irevcm, u, v, wgt, work);

  if (irevcm == -1)
    % v = A^Tu
    [v, ifail] = f11xa(...
                       'T', a(1:nz), irow(1:nz), icol(1:nz), 'N', u);
  elseif (irevcm == 1)
    % v = Au
    [v, ifail] = f11xa(...
                       'N', a(1:nz), irow(1:nz), icol(1:nz), 'N', u);
  elseif (irevcm == 2)
    % Solve ILU v = u  
    [v, ifail] = f11db('N', a, irow, icol, ipivp, ipivq, istr, idiag, 'N', u);
    if (ifail ~= 0)
      irevcm = 6;
    end
  elseif (irevcm == 3)
    % Monitoring stage
    [itn, stplhs, stprhs, anorm, sigmax, ifail] = ...
      f11bf(work);
    fprintf('\nMonitoring at iteration number %2d\n',itn);
    fprintf('residual norm:              %14.4e\n', stplhs);
    fprintf('\n   Solution Vector  Residual Vector\n');
    for i = 1:n
      fprintf('%16.4f %16.2e\n', u(i), v(i));
    end
  end
end

% Get information about the computation
[itn, stplhs, stprhs, anorm, sigmax, ifail] = ...
      f11bf(work);

fprintf('\nNumber of iterations for convergence:     %4d\n', itn);
fprintf('Residual norm:                           %14.4e\n', stplhs);
fprintf('Right-hand side of termination criteria: %14.4e\n', stprhs);
fprintf('i-norm of matrix a:                      %14.4e\n', anorm);
fprintf('\n   Solution Vector  Residual Vector\n');
for i = 1:n
  fprintf('%16.4f %16.2e\n', u(i), v(i));
end

f11bd example results


Monitoring at iteration number  1
residual norm:                  1.4059e+02

   Solution Vector  Residual Vector
         -4.5858         1.53e+01
          1.0154         2.66e+01
         -2.2234        -8.75e+00
          6.0097         1.86e+01
          1.3827         8.28e+00
         -7.9070         2.04e+01
          0.4427         9.61e+00
          5.9248         3.31e+01

Monitoring at iteration number  2
residual norm:                  3.2742e+01

   Solution Vector  Residual Vector
          4.1642        -2.96e+00
          4.9370        -5.55e+00
          4.8101         8.21e-01
          5.4324        -1.68e+01
          5.8531         5.60e-01
         11.9250        -1.91e+00
          8.4826         1.01e+00
          6.0625        -3.10e+00

Number of iterations for convergence:        3
Residual norm:                               1.0373e-08
Right-hand side of termination criteria:     5.8900e-06
i-norm of matrix a:                          1.1000e+01

   Solution Vector  Residual Vector
          1.0000        -1.36e-09
          2.0000        -2.61e-09
          3.0000         2.25e-10
          4.0000        -3.22e-09
          5.0000         6.30e-10
          6.0000        -5.24e-10
          7.0000         9.58e-10
          8.0000        -8.49e-10

Method	Requirements
RGMRES	$lwork = 100 + n (m + 3) + m (m + 5) + 1$ , where $m$ is the dimension of the basis.
CGS	$lwork = 100 + 7 n$ .
Bi-CGSTAB( $ℓ$ )	$lwork = 100 + (2 n + ℓ) (ℓ + 2) + p$ , where $ℓ$ is the order of the method.
TFQMR	$lwork = 100 + 10 n$ ,

$p = 2 n$	if $ℓ > 1$ and $iterm = 2$ was supplied.
$p = n$	if $ℓ > 1$ and a preconditioner is used or $iterm = 2$ was supplied.
$p = 0$	otherwise.

NAG Toolbox: nag_sparse_real_gen_basic_setup (f11bd)

▸▿ Contents

Purpose

Syntax

Description

Restarted Generalized Minimum Residual Method (RGMRES)

Conjugate Gradient Squared Method (CGS)

Bi-Conjugate Gradient Stabilized (ℓ) Method (Bi-CGSTAB(ℓ))

Transpose-free Quasi-minimal Residual Method (TFQMR)

General Considerations

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example