NAG Library Routine Document
F04ZCF
1 Purpose
F04ZCF estimates the $1$norm of a complex matrix without accessing the matrix explicitly. It uses reverse communication for evaluating matrixvector products. The routine may be used for estimating matrix condition numbers.
2 Specification
INTEGER 
ICASE, N, IFAIL 
REAL (KIND=nag_wp) 
ESTNRM 
COMPLEX (KIND=nag_wp) 
X(N), WORK(N) 

3 Description
F04ZCF computes an estimate (a lower bound) for the
$1$norm
of an
$n$ by
$n$ complex matrix
$A=\left({a}_{ij}\right)$. The routine regards the matrix
$A$ as being defined by a usersupplied ‘Black Box’ which, given an input vector
$x$, can return either of the matrixvector products
$Ax$ or
${A}^{\mathrm{H}}x$, where
${A}^{\mathrm{H}}$ is the complex conjugate transpose. A reverse communication interface is used; thus control is returned to the calling program whenever a matrixvector product is required.
Note: this routine is
not
recommended for use when the elements of
$A$ are known explicitly; it is then more efficient to compute the
$1$norm directly from the formula
(1) above.
The main
use of the routine is for estimating ${\Vert {B}^{1}\Vert}_{1}$, and hence the condition number
${\kappa}_{1}\left(B\right)={\Vert B\Vert}_{1}{\Vert {B}^{1}\Vert}_{1}$, without forming ${B}^{1}$ explicitly ($A={B}^{1}$ above).
If, for example, an $LU$ factorization of $B$ is available, the matrixvector products ${B}^{1}x$ and ${B}^{\mathrm{H}}x$ required by F04ZCF may be computed by back and forwardsubstitutions, without computing ${B}^{1}$.
The routine can also be used to estimate
$1$norms of matrix products such as
${A}^{1}B$ and
$ABC$, without forming the products explicitly. Further applications are described in
Higham (1988).
Since ${\Vert A\Vert}_{\infty}={\Vert {A}^{\mathrm{H}}\Vert}_{1}$, F04ZCF can be used to estimate the $\infty $norm of $A$ by working with ${A}^{\mathrm{H}}$ instead of $A$.
The algorithm used is based on a method given in
Hager (1984) and is described in
Higham (1988). A comparison of several techniques for condition number estimation is given in
Higham (1987).
Note: F04ZDF can also be used to estimate the norm of a real matrix.
F04ZDF uses a more recent algorithm than F04ZCF and it is recommended that
F04ZDF be used in place of F04ZCF.
4 References
Hager W W (1984) Condition estimates SIAM J. Sci. Statist. Comput. 5 311–316
Higham N J (1987) A survey of condition number estimation for triangular matrices SIAM Rev. 29 575–596
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
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 ICASE. Between intermediate exits and reentries,
all parameters other than X must remain unchanged.
 1: ICASE – INTEGERInput/Output
On initial entry: must be set to $0$.
On intermediate exit:
${\mathbf{ICASE}}=1$ or
$2$, and
${\mathbf{X}}\left(\mathit{i}\right)$, for
$\mathit{i}=1,2,\dots ,n$, contain the elements of a vector
$x$. The calling program must
(a) 
evaluate $Ax$ (if ${\mathbf{ICASE}}=1$) or ${A}^{\mathrm{H}}x$ (if ${\mathbf{ICASE}}=2$), where ${A}^{\mathrm{H}}$ is the complex conjugate transpose; 
(b) 
place the result in X; and, 
(c) 
call F04ZCF once again, with all the other parameters unchanged. 
On final exit: ${\mathbf{ICASE}}=0$.
 2: N – INTEGERInput
On initial entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 1$.
 3: X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate exit:
contains the current vector $x$.
On intermediate reentry: must contain $Ax$ (if ${\mathbf{ICASE}}=1$) or ${A}^{\mathrm{H}}x$ (if ${\mathbf{ICASE}}=2$).
On final exit: the array is undefined.
 4: ESTNRM – REAL (KIND=nag_wp)Input/Output
On initial entry: need not be set.
On intermediate exit:
should not be changed.
On final exit: an estimate (a lower bound) for ${\Vert A\Vert}_{1}$.
 5: WORK(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On final exit: contains a vector
$v$ such that
$v=Aw$ where
${\mathbf{ESTNRM}}={\Vert v\Vert}_{1}/{\Vert w\Vert}_{1}$ (
$w$ is not returned). If
$A={B}^{1}$ and
ESTNRM is large, then
$v$ is an approximate null vector for
$B$.
 6: IFAIL – INTEGERInput/Output

On 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, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On 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 entry,  ${\mathbf{N}}<1$. 
7 Accuracy
In extensive tests on
random matrices of size up to
$n=100$ the estimate
ESTNRM has been found always to be within a factor eleven of
${\Vert A\Vert}_{1}$; often the estimate has many correct figures. However, matrices exist for which the estimate is smaller than
${\Vert A\Vert}_{1}$ by an arbitrary factor; such matrices are very unlikely to arise in practice. See
Higham (1988) for further details.
The total time taken by F04ZCF is proportional to $n$. For most problems the time taken during calls to F04ZCF will be negligible compared with the time spent evaluating matrixvector products between calls to F04ZCF.
The number of matrixvector products required varies from $5$ to $11$ (or is $1$ if $n=1$). In most cases $5$ products are required; it is rare for more than $7$ to be needed.
It is your responsibility to guard against potential overflows during evaluation of the matrixvector products. In particular, when estimating ${\Vert {B}^{1}\Vert}_{1}$ using a triangular factorization of $B$, F04ZCF should not be called if one of the factors is exactly singular – otherwise division by zero may occur in the substitutions.
To estimate the
$1$norm of the inverse of a matrix
$A$, the following skeleton code can normally be used:
... code to factorize A ...
IF (A is not singular) THEN
ICASE = 0
10 CALL F04ZCF (ICASE,N,X,ESTNRM,WORK,IFAIL)
IF (ICASE.NE.0) THEN
IF (ICASE.EQ.1) THEN
... code to compute A(1)x ...
ELSE
... code to compute (A(1)(H)) x ...
END IF
GO TO 10
END IF
END IF
To compute ${A}^{1}x$ or ${A}^{\mathrm{H}}x$, solve the equation $Ay=x$ or ${A}^{\mathrm{H}}y=x$ for $y$, overwriting $y$ on $x$. The code will vary, depending on the type of the matrix $A$, and the NAG routine used to factorize $A$.
Note that if
$A$ is any type of
Hermitian matrix, then
$A={A}^{\mathrm{H}}$, and the
code following the call of F04ZCF
can be reduced to:
IF (ICASE.NE.0) THEN
... code to compute A(1)x ...
GO TO 10
END IF
The example program in
Section 9 illustrates how F04ZCF can be used in conjunction with NAG Library routines for complex band matrices (factorized by
F07BRF (ZGBTRF)).
It is also straightforward to use F04ZCF for Hermitian positive definite matrices, using
F06TFF,
F07FRF (ZPOTRF) and
F07FSF (ZPOTRS) for factorization and solution.
For upper or lower triangular matrices, no factorization routine is needed:
${A}^{1}x$ and
${A}^{\mathrm{H}}x$ may be computed by calls to
F06SJF (ZTRSV) (or
F06SKF (ZTBSV) if the matrix is banded, or
F06SLF (ZTPSV) if the matrix is stored in packed form).
9 Example
This example estimates the condition number
${\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}$ of the order
$5$ matrix
where
$A$ is a band matrix stored in the packed format required by
F07BRF (ZGBTRF) and
F07BSF (ZGBTRS).
Further examples of the technique for condition number estimation in the case of real matrices can be seen in the example program section of
F04YCF.
9.1 Program Text
Program Text (f04zcfe.f90)
9.2 Program Data
Program Data (f04zcfe.d)
9.3 Program Results
Program Results (f04zcfe.r)