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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_norm_rcomm (f04zc)

## Purpose

nag_linsys_complex_norm_rcomm (f04zc) estimates the 1$1$-norm of a complex matrix without accessing the matrix explicitly. It uses reverse communication for evaluating matrix-vector products. The function may be used for estimating matrix condition numbers.
Note: this function is scheduled to be withdrawn, please see f04zc in Advice on Replacement Calls for Withdrawn/Superseded Routines..

## Syntax

[icase, x, estnrm, work, ifail] = f04zc(icase, x, estnrm, work, 'n', n)
[icase, x, estnrm, work, ifail] = nag_linsys_complex_norm_rcomm(icase, x, estnrm, work, 'n', n)

## Description

nag_linsys_complex_norm_rcomm (f04zc) computes an estimate (a lower bound) for the 1$1$-norm
 n ‖A‖1 = max ∑ |aij| 1 ≤ j ≤ n i = 1
$‖A‖1 = max 1≤j≤n ∑ i=1 n |aij|$
(1)
of an n$n$ by n$n$ complex matrix A = (aij)$A=\left({a}_{ij}\right)$. The function regards the matrix A$A$ as being defined by a user-supplied ‘Black Box’ which, given an input vector x$x$, can return either of the matrix-vector products Ax$Ax$ or AHx${A}^{\mathrm{H}}x$, where AH${A}^{\mathrm{H}}$ is the complex conjugate transpose. A reverse communication interface is used; thus control is returned to the calling program whenever a matrix-vector product is required.
Note:  this function is not recommended for use when the elements of A$A$ are known explicitly; it is then more efficient to compute the 1$1$-norm directly from the formula (1) above.
The main use of the function is for estimating B11${‖{B}^{-1}‖}_{1}$, and hence the condition number κ1(B) = B1B11${\kappa }_{1}\left(B\right)={‖B‖}_{1}{‖{B}^{-1}‖}_{1}$, without forming B1${B}^{-1}$ explicitly (A = B1$A={B}^{-1}$ above).
If, for example, an LU$LU$ factorization of B$B$ is available, the matrix-vector products B1x${B}^{-1}x$ and BHx${B}^{-\mathrm{H}}x$ required by nag_linsys_complex_norm_rcomm (f04zc) may be computed by back- and forward-substitutions, without computing B1${B}^{-1}$.
The function can also be used to estimate 1$1$-norms of matrix products such as A1B${A}^{-1}B$ and ABC$ABC$, without forming the products explicitly. Further applications are described in Higham (1988).
Since A = AH1${‖A‖}_{\infty }={‖{A}^{\mathrm{H}}‖}_{1}$, nag_linsys_complex_norm_rcomm (f04zc) can be used to estimate the $\infty$-norm of A$A$ by working with AH${A}^{\mathrm{H}}$ instead of A$A$.
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: nag_linsys_complex_gen_norm_rcomm (f04zd) can also be used to estimate the norm of a real matrix. nag_linsys_complex_gen_norm_rcomm (f04zd) uses a more recent algorithm than nag_linsys_complex_norm_rcomm (f04zc) and it is recommended that nag_linsys_complex_gen_norm_rcomm (f04zd) be used in place of nag_linsys_complex_norm_rcomm (f04zc).

## 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 one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## Parameters

Note:  this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter icase. Between intermediate exits and re-entries, all parameters other than x must remain unchanged.

### Compulsory Input Parameters

1:     icase – int64int32nag_int scalar
On initial entry: must be set to 0$0$.
2:     x(n) – complex array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
On intermediate re-entry: must contain Ax$Ax$ (if icase = 1${\mathbf{icase}}=1$) or AHx${A}^{\mathrm{H}}x$ (if icase = 2${\mathbf{icase}}=2$).
3:     estnrm – double scalar
On initial entry: need not be set.
4:     work(n) – complex array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
On initial entry: need not be set.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the arrays x, work. (An error is raised if these dimensions are not equal.)
On initial entry: n$n$, the order of the matrix A$A$.
Constraint: n1${\mathbf{n}}\ge 1$.

None.

### Output Parameters

1:     icase – int64int32nag_int scalar
On intermediate exit: icase = 1${\mathbf{icase}}=1$ or 2$2$, and x(i)${\mathbf{x}}\left(\mathit{i}\right)$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, contain the elements of a vector x$x$. The calling program must
 (a) evaluate Ax$Ax$ (if icase = 1${\mathbf{icase}}=1$) or AHx${A}^{\mathrm{H}}x$ (if icase = 2${\mathbf{icase}}=2$), where AH${A}^{\mathrm{H}}$ is the complex conjugate transpose; (b) place the result in x; and, (c) call nag_linsys_complex_norm_rcomm (f04zc) once again, with all the other parameters unchanged.
On final exit: icase = 0${\mathbf{icase}}=0$.
2:     x(n) – complex array
On intermediate exit: contains the current vector x$x$.
On final exit: the array is undefined.
3:     estnrm – double scalar
On intermediate exit: should not be changed.
On final exit: an estimate (a lower bound) for A1${‖A‖}_{1}$.
4:     work(n) – complex array
On final exit: contains a vector v$v$ such that v = Aw$v=Aw$ where estnrm = v1 / w1${\mathbf{estnrm}}={‖v‖}_{1}/{‖w‖}_{1}$ (w$w$ is not returned). If A = B1$A={B}^{-1}$ and estnrm is large, then v$v$ is an approximate null vector for B$B$.
5:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$.

## Accuracy

In extensive tests on random matrices of size up to n = 100$n=100$ the estimate estnrm has been found always to be within a factor eleven of A1${‖A‖}_{1}$; often the estimate has many correct figures. However, matrices exist for which the estimate is smaller than A1${‖A‖}_{1}$ by an arbitrary factor; such matrices are very unlikely to arise in practice. See Higham (1988) for further details.

### Timing

The total time taken by nag_linsys_complex_norm_rcomm (f04zc) is proportional to n$n$. For most problems the time taken during calls to nag_linsys_complex_norm_rcomm (f04zc) will be negligible compared with the time spent evaluating matrix-vector products between calls to nag_linsys_complex_norm_rcomm (f04zc).
The number of matrix-vector products required varies from 5$5$ to 11$11$ (or is 1$1$ if n = 1$n=1$). In most cases 5$5$ products are required; it is rare for more than 7$7$ to be needed.

### Overflow

It is your responsibility to guard against potential overflows during evaluation of the matrix-vector products. In particular, when estimating B11${‖{B}^{-1}‖}_{1}$ using a triangular factorization of B$B$, nag_linsys_complex_norm_rcomm (f04zc) should not be called if one of the factors is exactly singular – otherwise division by zero may occur in the substitutions.

### Use in Conjunction with NAG Library Routines

To estimate the 1$1$-norm of the inverse of a matrix A$A$, the following skeleton code can normally be used:
...  code to factorize A ...
if (A is not singular)
icase = 0
[icase, x, estnrm, work, ifail] = f04zc(icase, x, estnrm, work);
while (icase ~= 0)
if (icase == 1)
...  code to compute A(-1)x ...
else
...  code to compute (A(-1)(H)) x ...
end
[icase, x, estnrm, work, ifail] = f04zc(icase, x, estnrm, work);
end
end
To compute A1x${A}^{-1}x$ or AHx${A}^{-\mathrm{H}}x$, solve the equation Ay = x$Ay=x$ or AHy = x${A}^{\mathrm{H}}y=x$ for y$y$, overwriting y$y$ on x$x$. The code will vary, depending on the type of the matrix A$A$, and the NAG function used to factorize A$A$.
Note that if A$A$ is any type of Hermitian matrix, then A = AH$A={A}^{\mathrm{H}}$, and the if statement after the while can be reduced to:
...  code to compute A(-1)x ...
The example program in Section [Example] illustrates how nag_linsys_complex_norm_rcomm (f04zc) can be used in conjunction with NAG Toolbox functions for complex band matrices (factorized by nag_lapack_zgbtrf (f07br)).
It is also straightforward to use nag_linsys_complex_norm_rcomm (f04zc) for Hermitian positive definite matrices, using nag_lapack_zpotrf (f07fr) and nag_lapack_zpotrs (f07fs) for factorization and solution.

## Example

function nag_linsys_complex_norm_rcomm_example
a = [ 1.0 + 1.0i,  2.0 + 1.0i,  1.0 + 2.0i,  0.0 + 0.0i,  0.0 + 0.0i;
0.0 + 2.0i,  3.0 + 5.0i,  1.0 + 3.0i,  2.0 + 1.0i,  0.0 + 0.0i,
0.0 + 0.0i,  -2.0 + 6.0i,  5.0 + 7.0i,  0.0 + 6.0i,  1.0 - 1.0i;
0.0 + 0.0i,  0.0 + 0.0i,  3.0 + 9.0i,  0.0 + 4.0i,  4.0 - 3.0i;
0.0 + 0.0i,  0.0 + 0.0i,  0.0 + 0.0i,  -1.0 + 8.0i,  10.0 - 3.0i];
icase = int64(0);
x = complex(zeros(5, 1));
estnrm = 0;
work = complex(zeros(5,1));
anorm = norm(a,1);
fprintf('Computed norm of a = %6.4g\n', anorm);
[icase, x, estnrm, work, ifail] = nag_linsys_complex_norm_rcomm(icase, x, estnrm, work);
while (icase ~= 0)
if (icase == 1)
x = inv(a)*x;
else
x = conj(transpose(inv(a)))*x;
end
[icase, x, estnrm, work, ifail] = nag_linsys_complex_norm_rcomm(icase, x, estnrm, work);
end
fprintf('Estimated norm of inverse(a) = %6.4g\n', estnrm);
fprintf('Estimated condition number of A = %6.1f\n', estnrm*anorm);

Computed norm of a =  23.49
Estimated norm of inverse(a) =  37.04
Estimated condition number of A =  870.0

function f04zc_example
a = [ 1.0 + 1.0i,  2.0 + 1.0i,  1.0 + 2.0i,  0.0 + 0.0i,  0.0 + 0.0i;
0.0 + 2.0i,  3.0 + 5.0i,  1.0 + 3.0i,  2.0 + 1.0i,  0.0 + 0.0i,
0.0 + 0.0i,  -2.0 + 6.0i,  5.0 + 7.0i,  0.0 + 6.0i,  1.0 - 1.0i;
0.0 + 0.0i,  0.0 + 0.0i,  3.0 + 9.0i,  0.0 + 4.0i,  4.0 - 3.0i;
0.0 + 0.0i,  0.0 + 0.0i,  0.0 + 0.0i,  -1.0 + 8.0i,  10.0 - 3.0i];
icase = int64(0);
x = complex(zeros(5, 1));
estnrm = 0;
work = complex(zeros(5,1));
anorm = norm(a,1);
fprintf('Computed norm of a = %6.4g\n', anorm);
[icase, x, estnrm, work, ifail] = f04zc(icase, x, estnrm, work);
while (icase ~= 0)
if (icase == 1)
x = inv(a)*x;
else
x = conj(transpose(inv(a)))*x;
end
[icase, x, estnrm, work, ifail] = f04zc(icase, x, estnrm, work);
end
fprintf('Estimated norm of inverse(a) = %6.4g\n', estnrm);
fprintf('Estimated condition number of A = %6.1f\n', estnrm*anorm);

Computed norm of a =  23.49
Estimated norm of inverse(a) =  37.04
Estimated condition number of A =  870.0