f11gef is an iterative solver for a symmetric system of simultaneous linear equations; f11gef is the second in a suite of three routines, where the first routine, f11gdf, must be called prior to f11gef to set up the suite, and the third routine in the suite, f11gff, can be used to return additional information about the computation.
These three routines are suitable for the solution of large sparse symmetric systems of equations.
For a general description of the methods employed you are referred to Section 3 in f11gdf.
f11gef can solve the system after the first routine in the suite, f11gdf, has been called to initialize the computation and specify the method of solution. The third routine in the suite, f11gff, can be used to return additional information generated by the computation during monitoring steps and after f11gef has completed its tasks.
f11gef uses reverse communication, i.e., f11gef returns repeatedly to the calling program with the argument irevcm (see Section 5) set to specified values which require the calling program to carry out a specific task: either to compute the matrix-vector 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 routine, 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 routines used to perform matrix operations.
4References
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 one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software14 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
5Arguments
Note: this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other thanirevcm and v must remain unchanged.
1: $\mathbf{irevcm}$ – IntegerInput/Output
On initial entry: ${\mathbf{irevcm}}=0$, otherwise an error condition will be raised.
On intermediate re-entry: 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. f11gef will then return with the termination code ${\mathbf{irevcm}}=4$. Note that before calling f11gef with ${\mathbf{irevcm}}=5$ the calling program must have performed the tasks required by the value of irevcm returned by the previous call to f11gef, otherwise subsequently returned values may be invalid.
${\mathbf{irevcm}}=6$
Immediate termination: f11gef 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. f11gef will then return with the termination code ${\mathbf{irevcm}}=4$.
Immediate termination may be useful, for example, when errors are detected during matrix-vector multiplication or during the solution of the preconditioning equation.
Changing irevcm to any other value between calls will result in an error.
On intermediate exit:
has the following meanings.
${\mathbf{irevcm}}=1$
The calling program must compute the matrix-vector 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 f11gff can be called at this step.
On final exit: if ${\mathbf{irevcm}}=4$, f11gef 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.
Constraint:
on initial entry, ${\mathbf{irevcm}}=0$; on re-entry, either irevcm must remain unchanged or be reset to ${\mathbf{irevcm}}=5$ or $6$.
Note: any values you return to f11gef as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by f11gef. If your code does inadvertently return any NaNs or infinities, f11gef is likely to produce unexpected results.
2: $\mathbf{u}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: 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 re-entry: must remain unchanged.
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 f11gef. If ${\mathbf{ifail}}={\mathbf{1}}$, the array u is unchanged from the last entry to f11gef. Otherwise, u holds the last iterate of the solution of the system of equations, for all returned values of ifail.
3: $\mathbf{v}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array v
must be at least
$\mathit{n}$.
On initial entry: the right-hand side $b$ of the system of equations $Ax=b$.
On intermediate re-entry: 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 f11gef.
If ${\mathbf{irevcm}}=3$, v must remain unchanged.
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 f11gef. If ${\mathbf{ifail}}={\mathbf{1}}$, the array v is unchanged from the initial entry to f11gef. If ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{2}}$, the array v contains the true residual vector of the system of equations (see also Section 6). 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: $\mathbf{wgt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array wgt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathit{n})$.
On entry: the user-supplied weights, if these are to be used in the computation of the vector norms in the termination criterion (see Sections 3 and 5 in f11gdf).
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: $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On initial entry: the array work as returned by f11gdf (see also Section 5 in f11gdf).
On intermediate re-entry: must remain unchanged.
6: $\mathbf{lwork}$ – IntegerInput
On initial entry: the dimension of the array work as declared in the (sub)program from which f11gef is called (see also Section 3 in f11gdf).
The required amount of workspace is as follows:
Method
Requirements
CG
${\mathbf{lwork}}=120+5\mathit{n}+p$.
SYMMLQ
${\mathbf{lwork}}=120+6\mathit{n}+p$,
MINRES
${\mathbf{lwork}}=120+9\mathit{n}$,
where
$p=2*({\mathbf{maxits}}+1)$, when an estimate of ${\sigma}_{1}\left(A\right)$ (sigmax) is computed;
$p=0$, otherwise.
Constraint:
${\mathbf{lwork}}\ge {\mathbf{lwreq}}$, where lwreq is returned by f11gdf.
7: $\mathbf{ifail}$ – IntegerInput/Output
On initial entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On final exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
f11gef has already completed its tasks. You need to set a new problem.
${\mathbf{ifail}}=2$
The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
User-requested termination: the required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
${\mathbf{ifail}}=3$
Either f11gdf was not called before calling this routine or it has returned an error.
${\mathbf{ifail}}=4$
User-requested tidy termination. The solution has not converged after $\u27e8\mathit{\text{value}}\u27e9$ iterations.
${\mathbf{ifail}}=5$
The solution has not converged after $\u27e8\mathit{\text{value}}\u27e9$ iterations.
${\mathbf{ifail}}=6$
The preconditioner appears not to be positive definite. The computation cannot continue.
${\mathbf{ifail}}=7$
The matrix of the coefficients $A$ appears not to be positive definite. The computation cannot continue.
${\mathbf{ifail}}=8$
User-requested immediate termination.
${\mathbf{ifail}}=9$
The matrix of the coefficients $A$ appears to be singular. The computation cannot continue.
On initial entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{irevcm}}=0$.
On intermediate re-entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: either irevcm must be unchanged from its previous exit value or ${\mathbf{irevcm}}=5$ or $6$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{lwork}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{lwork}}\ge {\mathbf{lwreq}}$, where lwreq is returned by f11gdf.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
On completion, i.e., ${\mathbf{irevcm}}=4$ on exit, the arrays u and v will return the solution and residual vectors, ${x}_{k}$ and ${r}_{k}=b-A{x}_{k}$, 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 user-specified tolerance, as described in Section 3 in f11gdf. The computed values of the left- and right-hand sides of the termination criterion selected can be obtained by a call to f11gff.
8Parallelism and Performance
f11gef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11gef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The number of operations carried out by f11gef for each iteration is likely to be principally determined by the computation of the matrix-vector 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 f11gef 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 $\overline{A}={E}^{-1}A{E}^{-\mathrm{T}}$.
Additional matrix-vector products are required for the computation of ${\Vert A\Vert}_{1}={\Vert A\Vert}_{\infty}$, when this has not been supplied to f11gdf and is required by the termination criterion employed.
The number of operations required to compute ${\sigma}_{1}\left(\overline{A}\right)$ is negligible for reasonable values of sigtol and maxits (see Sections 5 and 9 in f11gdf).
If the termination criterion ${\Vert {r}_{k}\Vert}_{p}\le \tau ({\Vert b\Vert}_{p}+{\Vert A\Vert}_{p}\times {\Vert {x}_{k}\Vert}_{p})$ is used (see Section 3 in f11gdf) 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: f11gef 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 ${x}_{0}=0$.