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NAG Toolbox: nag_sparse_real_symm_basic_solver (f11ge)
Purpose
nag_sparse_real_symm_basic_solver (f11ge) is an iterative solver for a symmetric system of simultaneous linear equations;
nag_sparse_real_symm_basic_solver (f11ge) is the second in a suite of three functions, where the first function,
nag_sparse_real_symm_basic_setup (f11gd), must be called prior to
nag_sparse_real_symm_basic_solver (f11ge) to set up the suite, and the third function in the suite,
nag_sparse_real_symm_basic_diag (f11gf), can be used to return additional information about the computation.
These three functions are suitable for the solution of large sparse symmetric systems of equations.
Syntax
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 22: 
lwork was removed from the interface 
Description
nag_sparse_real_symm_basic_solver (f11ge) solves the symmetric system of linear simultaneous equations
$Ax=b$ using the preconditioned conjugate gradient method (see
Hestenes and Stiefel (1952),
Golub and Van Loan (1996),
Barrett et al. (1994) and
Dias da Cunha and Hopkins (1994)), a preconditioned Lanczos method based upon the algorithm SYMMLQ (see
Paige and Saunders (1975) and
Barrett et al. (1994)), or the MINRES algorithm (see
Paige and Saunders (1975)).
For a general description of the methods employed you are referred to
Description in
nag_sparse_real_symm_basic_setup (f11gd).
nag_sparse_real_symm_basic_solver (f11ge) can solve the system after the first function in the suite,
nag_sparse_real_symm_basic_setup (f11gd), has been called to initialize the computation and specify the method of solution. The third function in the suite,
nag_sparse_real_symm_basic_diag (f11gf), can be used to return additional information generated by the computation during monitoring steps and after
nag_sparse_real_symm_basic_solver (f11ge) has completed its tasks.
nag_sparse_real_symm_basic_solver (f11ge) uses
reverse communication, i.e.,
nag_sparse_real_symm_basic_solver (f11ge) returns repeatedly to the calling program with the argument
irevcm (see
Arguments) set to specified values which require the calling program to carry out a specific task: either to compute the matrixvector product
$v=Au$; to solve the preconditioning equation
$Mv=u$; to notify the completion of the computation; or, to allow the calling program to monitor the solution. Through the argument
irevcm the calling program can cause immediate or tidy termination of the execution. On final exit, the last iterates of the solution and of the residual vectors of the original system of equations are returned.
Reverse communication has the following advantages.
1. 
Maximum flexibility in the representation and storage of sparse matrices: all matrix operations are performed outside the solver function, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns. This applies also to preconditioners. 
2. 
Enhanced user interaction: you can closely monitor the solution and tidy or immediate termination can be requested. This is useful, for example, when alternative termination criteria are to be employed or in case of failure of the external functions used to perform matrix operations. 
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
Dias da Cunha R and Hopkins T (1994) PIM 1.1 — the parallel iterative method package for systems of linear equations user's guide — Fortran 77 version Technical Report Computing Laboratory, University of Kent at Canterbury, Kent, UK
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
Higham N J (1988) FORTRAN codes for estimating the onenorm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Parameters
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and reentries,
all arguments other than irevcm and v must remain unchanged.
Compulsory Input Parameters
 1:
$\mathrm{irevcm}$ – int64int32nag_int scalar

On initial entry: ${\mathbf{irevcm}}=0$, otherwise an error condition will be raised.
On intermediate reentry: must either be unchanged from its previous exit value, or can have one of the following values.
 ${\mathbf{irevcm}}=5$
 Tidy termination: the computation will terminate at the end of the current iteration. Further reverse communication exits may occur depending on when the termination request is issued. nag_sparse_real_symm_basic_solver (f11ge) will then return with the termination code ${\mathbf{irevcm}}=4$. Note that before calling nag_sparse_real_symm_basic_solver (f11ge) with ${\mathbf{irevcm}}=5$ the calling program must have performed the tasks required by the value of irevcm returned by the previous call to nag_sparse_real_symm_basic_solver (f11ge), otherwise subsequently returned values may be invalid.
 ${\mathbf{irevcm}}=6$
 Immediate termination: nag_sparse_real_symm_basic_solver (f11ge) will return immediately with termination code ${\mathbf{irevcm}}=4$ and with any useful information available. This includes the last iterate of the solution and, for conjugate gradient only, the last iterate of the residual vector. The residual vector is generally not available when the Lanczos method (SYMMLQ) is used. nag_sparse_real_symm_basic_solver (f11ge) will then return with the termination code ${\mathbf{irevcm}}=4$.
Immediate termination may be useful, for example, when errors are detected during matrixvector multiplication or during the solution of the preconditioning equation.
Changing
irevcm to any other value between calls will result in an error.
Constraint:
on initial entry,
${\mathbf{irevcm}}=0$; on reentry, either
irevcm must remain unchanged or be reset to
${\mathbf{irevcm}}=5$ or
$6$.
 2:
$\mathrm{u}\left(:\right)$ – double array

The dimension of the array
u
must be at least
$\mathit{n}$
On initial entry: an initial estimate, ${x}_{0}$, of the solution of the system of equations $Ax=b$.
On intermediate reentry: must remain unchanged.
 3:
$\mathrm{v}\left(:\right)$ – double array

The dimension of the array
v
must be at least
$\mathit{n}$
On initial entry: the righthand side $b$ of the system of equations $Ax=b$.
On intermediate reentry: the returned value of
irevcm determines the contents of
v in the following way.
If
${\mathbf{irevcm}}=1$ or
$2$,
v must store the vector
$v$, the result of the operation specified by the value of
irevcm returned by the previous call to
nag_sparse_real_symm_basic_solver (f11ge).
If
${\mathbf{irevcm}}=3$,
v must remain unchanged.
 4:
$\mathrm{wgt}\left(:\right)$ – double array

The dimension of the array
wgt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{n}\right)$
The usersupplied weights, if these are to be used in the computation of the vector norms in the termination criterion (see
Description and
Arguments in
nag_sparse_real_symm_basic_setup (f11gd)).
Weights are NOT used in the MINRES algorithm.
Constraint:
if weights are to be used, at least one element of
wgt must be nonzero.
 5:
$\mathrm{work}\left(\mathit{lwork}\right)$ – double array

lwork, the dimension of the array, must satisfy the constraint
$\mathit{lwork}\ge {\mathbf{lwreq}}$, where
lwreq is returned by
nag_sparse_real_symm_basic_setup (f11gd).
On initial entry: the array
work as returned by
nag_sparse_real_symm_basic_setup (f11gd) (see also
Arguments in
nag_sparse_real_symm_basic_setup (f11gd)).
lwork, the dimension of the array, must satisfy the constraint
$\mathit{lwork}\ge {\mathbf{lwreq}}$, where
lwreq is returned by
nag_sparse_real_symm_basic_setup (f11gd).
On intermediate reentry: must remain unchanged.
Optional Input Parameters
None.
Output Parameters
 1:
$\mathrm{irevcm}$ – int64int32nag_int scalar

On intermediate exit:
has the following meanings.
 ${\mathbf{irevcm}}=1$
 The calling program must compute the matrixvector product $v=Au$, where $u$ and $v$ are stored in u and v, respectively.
 ${\mathbf{irevcm}}=2$
 The calling program must solve the preconditioning equation $Mv=u$, where $u$ and $v$ are stored in u and v, respectively.
 ${\mathbf{irevcm}}=3$
 Monitoring step: the solution and residual at the current iteration are returned in the arrays u and v, respectively. No action by the calling program is required. To return additional information nag_sparse_real_symm_basic_diag (f11gf) can be called at this step.
On final exit: if
${\mathbf{irevcm}}=4$,
nag_sparse_real_symm_basic_solver (f11ge) has completed its tasks. The value of
ifail determines whether the iteration has been successfully completed, errors have been detected or the calling program has requested termination.
 2:
$\mathrm{u}\left(:\right)$ – double array

The dimension of the array
u will be
$\mathit{n}$
On intermediate exit:
the returned value of
irevcm determines the contents of
u in the following way.
If
${\mathbf{irevcm}}=1$ or
$2$,
u holds the vector
$u$ on which the operation specified by
irevcm is to be carried out.
If
${\mathbf{irevcm}}=3$,
u holds the current iterate of the solution vector.
On final exit: if
${\mathbf{ifail}}={\mathbf{3}}$ or
${{\mathit{i}}}$, the array
u is unchanged from the initial entry to
nag_sparse_real_symm_basic_solver (f11ge). If
${\mathbf{ifail}}={\mathbf{1}}$, the array
u is unchanged from the last entry to
nag_sparse_real_symm_basic_solver (f11ge). Otherwise,
u holds the last iterate of the solution of the system of equations, for all returned values of
ifail.
 3:
$\mathrm{v}\left(:\right)$ – double array

The dimension of the array
v will be
$\mathit{n}$
On intermediate exit:
if
${\mathbf{irevcm}}=3$,
v holds the current iterate of the residual vector. Note that this is an approximation to the true residual vector. Otherwise, it does not contain any useful information.
On final exit: if
${\mathbf{ifail}}={\mathbf{3}}$ or
${{\mathit{i}}}$, the array
v is unchanged from the last entry to
nag_sparse_real_symm_basic_solver (f11ge). If
${\mathbf{ifail}}={\mathbf{1}}$, the array
v is unchanged from the initial entry to
nag_sparse_real_symm_basic_solver (f11ge). If
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{2}}$, the array
v contains the true residual vector of the system of equations (see also
Error Indicators and Warnings). Otherwise,
v stores the last iterate of the residual vector unless the Lanczos method (SYMMLQ) was used and
${\mathbf{ifail}}\ge {\mathbf{5}}$, in which case
v is set to
$0.0$.
 4:
$\mathrm{work}\left(\mathit{lwork}\right)$ – double array

 5:
$\mathrm{ifail}$ – int64int32nag_int scalar
On final exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
 W ${\mathbf{ifail}}=1$

nag_sparse_real_symm_basic_solver (f11ge) has already completed its tasks. You need to set a new problem.
 W ${\mathbf{ifail}}=2$

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
Userrequested termination: the required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
 ${\mathbf{ifail}}=3$

Either
nag_sparse_real_symm_basic_setup (f11gd) was not called before calling this function or it has returned an error.
 W ${\mathbf{ifail}}=4$

Userrequested tidy termination.
 W ${\mathbf{ifail}}=5$

The solution has not converged after $\_$ iterations.
 ${\mathbf{ifail}}=6$

The preconditioner appears not to be positive definite. The computation cannot continue.
 W ${\mathbf{ifail}}=7$

The matrix of the coefficients $A$ appears not to be positive definite. The computation cannot continue.
 W ${\mathbf{ifail}}=8$

Userrequested immediate termination.
 ${\mathbf{ifail}}=9$

The matrix of the coefficients $A$ appears to be singular. The computation cannot continue.
 ${\mathbf{ifail}}=10$

The weights in array
wgt are all zero.
 ${\mathbf{ifail}}=1$

On initial entry, ${\mathbf{irevcm}}=\_$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate reentry,
${\mathbf{irevcm}}=\_$.
Constraint: either
irevcm must be unchanged from its previous exit value or
${\mathbf{irevcm}}=5$ or
$6$.
 ${\mathbf{ifail}}=6$

Constraint:
$\mathit{lwork}\ge {\mathbf{lwreq}}$, where
lwreq is returned by
nag_sparse_real_symm_basic_setup (f11gd).
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
Accuracy
On completion, i.e.,
${\mathbf{irevcm}}=4$ on exit, the arrays
u and
v will return the solution and residual vectors,
${x}_{k}$ and
${r}_{k}=bA{x}_{k}$, respectively, at the
$k$th iteration, the last iteration performed, unless an immediate termination was requested and the Lanczos method (SYMMLQ) was used.
On successful completion, the termination criterion is satisfied to within the userspecified tolerance, as described in
Description in
nag_sparse_real_symm_basic_setup (f11gd). The computed values of the left and righthand sides of the termination criterion selected can be obtained by a call to
nag_sparse_real_symm_basic_diag (f11gf).
Further Comments
The number of operations carried out by nag_sparse_real_symm_basic_solver (f11ge) for each iteration is likely to be principally determined by the computation of the matrixvector products $v=Au$ and by the solution of the preconditioning equation $Mv=u$ in the calling program. Each of these operations is carried out once every iteration.
The number of the remaining operations in nag_sparse_real_symm_basic_solver (f11ge) for each iteration is approximately proportional to $\mathit{n}$. Note that the Lanczos method (SYMMLQ) requires a slightly larger number of operations than the conjugate gradient method.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined at the onset, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients $\stackrel{}{A}={E}^{1}A{E}^{\mathrm{T}}$.
Additional matrixvector products are required for the computation of
${\Vert A\Vert}_{1}={\Vert A\Vert}_{\infty}$, when this has not been supplied to
nag_sparse_real_symm_basic_setup (f11gd) and is required by the termination criterion employed.
The number of operations required to compute
${\sigma}_{1}\left(\stackrel{}{A}\right)$ is negligible for reasonable values of
sigtol and
maxits (see
Arguments and
Further Comments in
nag_sparse_real_symm_basic_setup (f11gd)).
If the termination criterion
${\Vert {r}_{k}\Vert}_{p}\le \tau \left({\Vert b\Vert}_{p}+{\Vert A\Vert}_{p}\times {\Vert {x}_{k}\Vert}_{p}\right)$ is used (see
Description in
nag_sparse_real_symm_basic_setup (f11gd)) and
$\Vert {x}_{0}\Vert \gg \Vert {x}_{k}\Vert $, so that because of loss of significant digits the required accuracy could not be obtained, the iteration is restarted automatically at some suitable point:
nag_sparse_real_symm_basic_solver (f11ge) sets
${x}_{0}={x}_{k}$ and the computation begins again. For particularly badly scaled problems, more than one restart may be necessary. Naturally, restarting adds to computational costs: it is recommended that the iteration should start from a value
${x}_{0}$ which is as close to the true solution
$\stackrel{~}{x}$ as can be estimated. Otherwise, the iteration should start from
${x}_{0}=0$.
Example
See
Example in
nag_sparse_real_symm_basic_setup (f11gd).
Open in the MATLAB editor:
f11ge_example
function f11ge_example
fprintf('f11ge example results\n\n');
n = int64(7);
nz = int64(16);
a = zeros(1000,1);
irow = zeros(1000,1,'int64');
icol = irow;
a(1:16) = [4 1 5 2 2 3 1 1 4 1 2 3 2 1 2 5];
irow(1:16) = [1 2 2 3 4 4 5 5 5 6 6 6 7 7 7 7];
icol(1:16) = [1 1 2 3 2 4 1 4 5 2 5 6 1 2 3 7];
lfill = int64(0);
dtol = 0;
mic = 'N';
dscale = 0;
ipiv = zeros(n, 1, 'int64');
[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = ...
f11ja( ...
nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv);
method = 'CG';
precon = 'P';
tol = 1e6;
maxitn = int64(20);
anorm = 0;
sigmax = 0;
maxits = int64(7);
monit = int64(2);
[lwreq, work, ifail] = ...
f11gd(method, precon, n, tol, maxitn, anorm, sigmax, ...
maxits, monit, 'sigcmp', 'S', 'norm_p', '1');
irevcm = int64(0);
u = zeros(n,1);
v = [15; 18; 8; 21; 11; 10; 29];
wgt = zeros(n,1);
while (irevcm ~= 4)
[irevcm, u, v, work, ifail] = f11ge( ...
irevcm, u, v, wgt, work);
if (irevcm == 1)
[v, ifail] = f11xe( ...
a, irow, icol, 'N', u, 'nz', nz);
elseif (irevcm == 2)
[v, ifail] = f11jb( ...
a, irow, icol, ipiv, istr, 'N', u);
elseif (irevcm == 3)
[itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = ...
f11gf(work);
fprintf('\nMonitoring at iteration number %d\n',itn);
fprintf('residual norm: %14.4e\n\n', stplhs);
fprintf(' Solution Vector Residual Vector\n');
fprintf('%16.4f %16.4e\n', [u'; v']);
end
end
[itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = f11gf(work);
fprintf('\nNumber of iterations for convergence: %d\n', itn);
fprintf('Residual norm: %14.4e\n', stplhs);
fprintf('Righthand side of termination criteria: %14.4e\n', stprhs);
fprintf('inorm of matrix a: %14.4e\n', anorm);
fprintf('\n Solution Vector Residual Vector\n');
fprintf('%16.4f %12.2e\n', [u'; v']);
f11ge example results
Monitoring at iteration number 2
residual norm: 1.9938e+00
Solution Vector Residual Vector
0.9632 2.2960e01
1.9934 2.2254e01
3.0583 9.5827e02
4.1453 2.5155e01
4.8289 1.7160e01
5.6630 6.7533e01
7.1062 3.4737e01
Monitoring at iteration number 4
residual norm: 6.6574e03
Solution Vector Residual Vector
0.9994 1.0551e03
2.0011 2.4675e03
3.0008 1.7116e05
3.9996 4.4929e05
4.9991 2.1359e03
5.9993 8.7482e04
7.0007 6.2045e05
Number of iterations for convergence: 5
Residual norm: 2.0428e14
Righthand side of termination criteria: 3.9200e04
inorm of matrix a: 1.0000e+01
Solution Vector Residual Vector
1.0000 0.00e+00
2.0000 0.00e+00
3.0000 2.66e15
4.0000 3.55e15
5.0000 5.33e15
6.0000 1.78e15
7.0000 7.11e15
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