NAG Library Routine Document
F12FBF
Note: this routine uses optional parameters to define choices in the problem specification. If you wish to use default
settings for all of the optional parameters, then the option setting routine
F12FDF
need not be called.
If, however, you wish to reset some or all of the settings please refer to
Section 10 in F12FDF
for a detailed description of the specification of the optional parameters.
1 Purpose
F12FBF is an iterative solver in a suite of routines consisting of
F12FAF, F12FBF,
F12FCF,
F12FDF and
F12FEF. It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real symmetric matrices.
2 Specification
INTEGER 
IREVCM, LDV, NSHIFT, ICOMM(*), IFAIL 
REAL (KIND=nag_wp) 
RESID(*), V(LDV,*), X(*), MX(*), COMM(*) 

3 Description
The suite of routines is designed to calculate some of the eigenvalues, $\lambda $, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, real and symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.
F12FBF is a
reverse communication routine, based on the ARPACK routine
dsaupd, using the Implicitly Restarted Arnoldi iteration method, which for symmetric problems reduces to a variant of the Lanczos method. The method is described in
Lehoucq and Sorensen (1996) and
Lehoucq (2001) while its use within the ARPACK software is described in great detail in
Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse symmetric matrices is provided in
Lehoucq and Scott (1996). This suite of routines offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify the interface of F12FBF.
The setup routine
F12FAF must be called before F12FBF, the reverse communication iterative solver. Options may be set for F12FBF by prior calls to the option setting routine
F12FDF and a postprocessing routine
F12FCF must be called following a successful final exit from F12FBF.
F12FEF, may be called following certain flagged, intermediate exits from F12FBF to provide additional monitoring information about the computation.
F12FBF uses
reverse communication, i.e., it returns repeatedly to the calling program with the parameter
IREVCM (see
Section 5) set to specified values which require the calling program to carry out one of the following tasks:
– 
compute the matrixvector product $y=\mathrm{OP}x$, where $\mathrm{OP}$ is defined by the computational mode; 
– 
compute the matrixvector product $y=Bx$; 
– 
notify the completion of the computation; 
– 
allow the calling program to monitor the solution. 
The problem type to be solved (standard or generalized), the spectrum of eigenvalues of interest, the mode used (regular, regular inverse, shifted inverse, Buckling or Cayley) and other options can all be set using the option setting routine
F12FDF.
4 References
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCSP5471195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5 Parameters
Note: this routine uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the parameter
IREVCM. Between intermediate exits and reentries,
all parameters other than X, MX and COMM must remain unchanged.
 1: IREVCM – INTEGERInput/Output
On initial entry: ${\mathbf{IREVCM}}=0$, otherwise an error condition will be raised.
On intermediate reentry: must be unchanged from its previous exit value. 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 matrixvector product $y=\mathrm{OP}x$, where $x$ is stored in X (by default) or in the array COMM (starting from the location given by the first element of ICOMM) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set in a prior call to F12FDF. The result $y$ is returned in X (by default) or in the array COMM (starting from the location given by the second element of ICOMM) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
 ${\mathbf{IREVCM}}=1$
 The calling program must compute the matrixvector product $y=\mathrm{OP}x$. This is similar to the case ${\mathbf{IREVCM}}=1$ except that the result of the matrixvector product $Bx$ (as required in some computational modes) has already been computed and is available in MX (by default) or in the array COMM (starting from the location given by the third element of ICOMM) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
 ${\mathbf{IREVCM}}=2$
 The calling program must compute the matrixvector product $y=Bx$, where $x$ is stored in X and $y$ is returned in MX (by default) or in the array COMM (starting from the location given by the second element of ICOMM) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
 ${\mathbf{IREVCM}}=3$
 Compute the NSHIFT real and imaginary parts of the shifts where the real parts are to be returned in the first NSHIFT locations of the array X and the imaginary parts are to be returned in the first NSHIFT locations of the array MX. Only complex conjugate pairs of shifts may be applied and the pairs must be placed in consecutive locations. This value of IREVCM will only arise if the optional parameter Supplied Shifts is set in a prior call to F12FDF which is intended for experienced users only; the default and recommended option is to use exact shifts (see Lehoucq et al. (1998) for details and guidance on the choice of shift strategies).
 ${\mathbf{IREVCM}}=4$
 Monitoring step: a call to F12FEF can now be made to return the number of Arnoldi iterations, the number of converged Ritz values, their real and imaginary parts, and the corresponding Ritz estimates.
On final exit:
${\mathbf{IREVCM}}=5$: F12FBF has completed its tasks. The value of
IFAIL determines whether the iteration has been successfully completed, or whether errors have been detected. On successful completion
F12FCF must be called to return the requested eigenvalues and eigenvectors (and/or Schur vectors).
Constraint:
on initial entry,
${\mathbf{IREVCM}}=0$; on reentry
IREVCM must remain unchanged.
 2: RESID($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
RESID
must be at least
${\mathbf{N}}$ (see
F12FAF).
On initial entry: need not be set unless the option
Initial Residual has been set in a prior call to
F12FDF in which case
RESID should contain an initial residual vector, possibly from a previous run.
On intermediate reentry: must be unchanged from its previous exit. Changing
RESID to any other value between calls may result in an error exit.
On intermediate exit:
contains the current residual vector.
On final exit: contains the final residual vector.
 3: V(LDV,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
V
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NCV}}\right)$ (see
F12FAF).
On initial entry: need not be set.
On intermediate reentry: must be unchanged from its previous exit.
On intermediate exit:
contains the current set of Arnoldi basis vectors.
On final exit: contains the final set of Arnoldi basis vectors.
 4: LDV – INTEGERInput
On entry: the first dimension of the array
V as declared in the (sub)program from which F12FBF is called.
Constraint:
${\mathbf{LDV}}\ge {\mathbf{N}}$.
 5: X($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array
X
must be at least
${\mathbf{N}}$ if
${\mathbf{Pointers}}=\mathrm{NO}$ (default) and at least
$1$ if
${\mathbf{Pointers}}=\mathrm{YES}$ (see
F12FAF).
On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of
COMM.
On intermediate reentry: if
${\mathbf{Pointers}}=\mathrm{YES}$,
X need not be set.
If
${\mathbf{Pointers}}=\mathrm{NO}$,
X must contain the result of
$y=\mathrm{OP}x$ when
IREVCM returns the value
$1$ or
$+1$. It must return the real parts of the computed shifts when
IREVCM returns the value
$3$.
On intermediate exit:
if
${\mathbf{Pointers}}=\mathrm{YES}$,
X is not referenced.
If
${\mathbf{Pointers}}=\mathrm{NO}$,
X contains the vector
$x$ when
IREVCM returns the value
$1$ or
$+1$.
On final exit: does not contain useful data.
 6: MX($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array
MX
must be at least
${\mathbf{N}}$ if
${\mathbf{Pointers}}=\mathrm{NO}$ (default) and at least
$1$ if
${\mathbf{Pointers}}=\mathrm{YES}$ (see
F12FAF).
On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of
COMM.
On intermediate reentry: if
${\mathbf{Pointers}}=\mathrm{YES}$,
MX need not be set.
If
${\mathbf{Pointers}}=\mathrm{NO}$,
MX must contain the result of
$y=Bx$ when
IREVCM returns the value
$2$. It must return the imaginary parts of the computed shifts when
IREVCM returns the value
$3$.
On intermediate exit:
if
${\mathbf{Pointers}}=\mathrm{YES}$,
MX is not referenced.
If
${\mathbf{Pointers}}=\mathrm{NO}$,
MX contains the vector
$Bx$ when
IREVCM returns the value
$+1$.
On final exit: does not contain any useful data.
 7: NSHIFT – INTEGEROutput
On intermediate exit:
if the option
Supplied Shifts is set and
IREVCM returns a value of
$3$,
NSHIFT returns the number of complex shifts required.
 8: COMM($*$) – REAL (KIND=nag_wp) arrayCommunication Array

Note: the dimension of the array
COMM
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LCOMM}}\right)$ (see
F12FAF).
On initial entry: must remain unchanged following a call to the setup routine
F12FAF.
On exit: contains data defining the current state of the iterative process.
 9: ICOMM($*$) – INTEGER arrayCommunication Array

Note: the dimension of the array
ICOMM
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LICOMM}}\right)$ (see
F12FAF).
On initial entry: must remain unchanged following a call to the setup routine
F12FAF.
On exit: contains data defining the current state of the iterative process.
 10: IFAIL – INTEGERInput/Output
On initial entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On initial entry, the maximum number of iterations
$\le 0$, the option
Iteration Limit has been set to a nonpositive value.
 ${\mathbf{IFAIL}}=2$
The options
Generalized and
Regular are incompatible.
 ${\mathbf{IFAIL}}=3$
Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues requested is one.
 ${\mathbf{IFAIL}}=4$
The option
Initial Residual was selected but the starting vector held in
RESID is zero.
 ${\mathbf{IFAIL}}=5$
The maximum number of iterations has been reached. Some Ritz values may have converged; a subsequent call to
F12FCF will return the number of converged values and the converged values.
 ${\mathbf{IFAIL}}=6$
No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration. One possibility is to increase the size of
NCV relative to
NEV (see
Section 5 in F12FAF for details of these parameters).
 ${\mathbf{IFAIL}}=7$
Could not build a Lanczos factorization. Consider changing
NCV or
NEV in the initialization routine (see
Section 5 in F12FAF for details of these parameters).
 ${\mathbf{IFAIL}}=8$
Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=9$
An unexpected error has occurred. Please contact
NAG.
7 Accuracy
The relative accuracy of a Ritz value,
$\lambda $, is considered acceptable if its Ritz estimate
$\text{}\le {\mathbf{Tolerance}}\times \left\lambda \right$. The default
Tolerance used is the
machine precision given by
X02AJF.
None.
9 Example
For this routine two examples are presented, with a main program and two example problems given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
The example solves $Ax=\lambda x$ in shiftinvert mode, where $A$ is obtained from the standard central difference discretization of the onedimensional Laplacian operator $\frac{{\partial}^{2}u}{\partial {x}^{2}}$ with zero Dirichlet boundary conditions. Eigenvalues closest to the shift $\sigma =0$ are sought.
Example 2 (EX2)
This example illustrates the use of F12FBF to compute the leading terms in the singular value decomposition of a real general matrix
$A$. The example finds a few of the largest singular values (
$\sigma $) and corresponding right singular values (
$\nu $) for the matrix
$A$ by solving the symmetric problem:
Here
$A$ is the
$m$ by
$n$ real matrix derived from the simplest finite difference discretization of the twodimensional kernal
$k\left(s,t\right)dt$ where
Note: this formulation is appropriate for the case $m\ge n$. Reverse the rules of $A$ and ${A}^{\mathrm{T}}$ in the case of $m<n$.
9.1 Program Text
Program Text (f12fbfe.f90)
9.2 Program Data
Program Data (f12fbfe.d)
9.3 Program Results
Program Results (f12fbfe.r)