Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_gen_norm_rcomm (f04zd)

## Purpose

nag_linsys_complex_gen_norm_rcomm (f04zd) estimates the 1$1$-norm of a complex rectangular matrix without accessing the matrix explicitly. It uses reverse communication for evaluating matrix products. The function may be used for estimating condition numbers of square matrices.

## Syntax

[irevcm, x, y, estnrm, work, rwork, iwork, ifail] = f04zd(irevcm, x, y, estnrm, seed, work, rwork, iwork, 'm', m, 'n', n, 't', t)
[irevcm, x, y, estnrm, work, rwork, iwork, ifail] = nag_linsys_complex_gen_norm_rcomm(irevcm, x, y, estnrm, seed, work, rwork, iwork, 'm', m, 'n', n, 't', t)

## Description

nag_linsys_complex_gen_norm_rcomm (f04zd) computes an estimate (a lower bound) for the 1$1$-norm
 m ‖A‖1 = max ∑ |aij| 1 ≤ j ≤ n i = 1
$‖A‖1 = max 1≤j≤n ∑ i=1 m |aij|$
(1)
of an m$m$ 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 n × t$n×t$ matrix X$X$ (with tn$t\ll n$) or an m × t$m×t$ matrix Y$Y$, can return AX$AX$ or AHY${A}^{\mathrm{H}}Y$, 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 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${‖{B}^{-1}‖}_{1}$ for a square matrix B$B$, 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 products B1X${B}^{-1}X$ and BHY${B}^{-\mathrm{H}}Y$ required by nag_linsys_complex_gen_norm_rcomm (f04zd) 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_gen_norm_rcomm (f04zd) 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 described in Higham and Tisseur (2000).

## References

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
Higham N J and Tisseur F (2000) A block algorithm for matrix 1$1$-norm estimation, with an application to 1$1$-norm pseudospectra SIAM J. Matrix. Anal. Appl. 21 1185–1201

## 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 irevcm. Between intermediate exits and re-entries, all parameters other than x and y must remain unchanged.

### Compulsory Input Parameters

1:     irevcm – int64int32nag_int scalar
On initial entry: must be set to 0$0$.
On intermediate re-entry: irevcm must be unchanged.
2:     x(ldx, : $:$) – complex array
The first dimension of the array x must be at least n${\mathbf{n}}$
The second dimension of the array must be at least max (1,t)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 2${\mathbf{irevcm}}=2$, must contain AHY${A}^{\mathrm{H}}Y$.
3:     y(ldy, : $:$) – complex array
The first dimension of the array y must be at least m${\mathbf{m}}$
The second dimension of the array must be at least max (1,t)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 1${\mathbf{irevcm}}=1$, must contain AX$AX$.
4:     estnrm – double scalar
On initial entry: need not be set.
On intermediate re-entry: must not be changed.
5:     seed – int64int32nag_int scalar
The seed used for random number generation.
If t = 1${\mathbf{t}}=1$, seed is not used.
6:     work(m × t${\mathbf{m}}×{\mathbf{t}}$) – complex array
7:     rwork(2 × n$2×{\mathbf{n}}$) – double array
8:     iwork(2 × n + 5 × t + 20$2×{\mathbf{n}}+5×{\mathbf{t}}+20$) – int64int32nag_int array
On initial entry: need not be set.
On intermediate re-entry: must not be changed.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the arrays y, work. (An error is raised if these dimensions are not equal.)
The number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The first dimension of the array x.
On initial entry: n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     t – int64int32nag_int scalar
Default: The second dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
The number of columns t$t$ of the matrices X$X$ and Y$Y$. This is a parameter that can be used to control the accuracy and reliability of the estimate and corresponds roughly to the number of columns of A$A$ that are visited during each iteration of the algorithm.
If t2${\mathbf{t}}\ge 2$ then a partly random starting matrix is used in the algorithm.
Default: t = 2${\mathbf{t}}=2$
Constraint: 1tm$1\le {\mathbf{t}}\le {\mathbf{m}}$.

ldx ldy

### Output Parameters

1:     irevcm – int64int32nag_int scalar
On intermediate exit: irevcm = 1${\mathbf{irevcm}}=1$ or 2$2$, and x and y contain the elements of n × t$n×t$ and m × t$m×t$ matrices respectively. The calling program must
 (a) if irevcm = 1${\mathbf{irevcm}}=1$, evaluate AX$AX$ and store the result in y or if irevcm = 2${\mathbf{irevcm}}=2$, evaluate AHY${A}^{\mathrm{H}}Y$ and store the result in x, where AH${A}^{\mathrm{H}}$ is the complex conjugate transpose; (b) call nag_linsys_complex_gen_norm_rcomm (f04zd) once again, with all the parameters unchanged.
On final exit: irevcm = 0${\mathbf{irevcm}}=0$.
2:     x(ldx, : $:$) – complex array
The first dimension of the array x will be n${\mathbf{n}}$
The second dimension of the array will be max (1,t)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$
ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
On intermediate exit: if irevcm = 1${\mathbf{irevcm}}=1$, contains the current matrix X$X$.
ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
On final exit: the array is undefined.
3:     y(ldy, : $:$) – complex array
The first dimension of the array y will be m${\mathbf{m}}$
The second dimension of the array will be max (1,t)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$
ldym$\mathit{ldy}\ge {\mathbf{m}}$.
On intermediate exit: if irevcm = 2${\mathbf{irevcm}}=2$, contains the current matrix Y$Y$.
ldym$\mathit{ldy}\ge {\mathbf{m}}$.
On final exit: the array is undefined.
4:     estnrm – double scalar
On final exit: an estimate (a lower bound) for A1${‖A‖}_{1}$.
5:     work(m × t${\mathbf{m}}×{\mathbf{t}}$) – complex array
6:     rwork(2 × n$2×{\mathbf{n}}$) – double array
7:     iwork(2 × n + 5 × t + 20$2×{\mathbf{n}}+5×{\mathbf{t}}+20$) – int64int32nag_int array
8:     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${\mathbf{ifail}}=1$
ifail = 1${\mathbf{ifail}}=-1$
Constraint: irevcm = 0${\mathbf{irevcm}}=0$, 1$1$ or 2$2$.
On initial entry.
Constraint: irevcm = 0${\mathbf{irevcm}}=0$.
ifail = 2${\mathbf{ifail}}=-2$
Constraint: m0${\mathbf{m}}\ge 0$.
ifail = 3${\mathbf{ifail}}=-3$
Constraint: n0${\mathbf{n}}\ge 0$.
ifail = 5${\mathbf{ifail}}=-5$
Constraint: ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
ifail = 7${\mathbf{ifail}}=-7$
Constraint: ldym$\mathit{ldy}\ge {\mathbf{m}}$.
ifail = 9${\mathbf{ifail}}=-9$
Constraint: 1tm$1\le {\mathbf{t}}\le {\mathbf{m}}$.

## Accuracy

In extensive tests on random matrices of size up to m = n = 450$m=n=450$ the estimate estnrm has been found always to be within a factor two 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 and Tisseur (2000) for further details.

### Timing

For most problems the time taken during calls to nag_linsys_complex_gen_norm_rcomm (f04zd) will be negligible compared with the time spent evaluating matrix products between calls to nag_linsys_complex_gen_norm_rcomm (f04zd).
The number of matrix products required depends on the matrix A$A$. At most six products of the form Y = AX$Y=AX$ and five products of the form X = AHY$X={A}^{\mathrm{H}}Y$ will be required. The number of iterations is independent of the choice of t$t$.

### Overflow

It is your responsibility to guard against potential overflows during evaluation of the matrix products. In particular, when estimating B11${‖{B}^{-1}‖}_{1}$ using a triangular factorization of B$B$, nag_linsys_complex_gen_norm_rcomm (f04zd) should not be called if one of the factors is exactly singular – otherwise division by zero may occur in the substitutions.

### Choice of t

The parameter t$t$ controls the accuracy and reliability of the estimate. For t = 1$t=1$, the algorithm behaves similarly to the LAPACK estimator xLACON. Increasing t$t$ typically improves the estimate, without increasing the number of iterations required.
For t2$t\ge 2$, random matrices are used in the algorithm, so for repeatable results the value of seed should be kept constant.
A value of t = 2$t=2$ is recommended for new users.

### 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] = f04zd(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] = f04zd(icase, x, estnrm, work);
end
end
```
To compute A1X${A}^{-1}X$ or AHY${A}^{-\mathrm{H}}Y$, solve the equation AY = X$AY=X$ or AHX = Y${A}^{\mathrm{H}}X=Y$ storing the result in y or x respectively. The code will vary, depending on the type of the matrix A$A$, and the NAG function used to factorize A$A$.
The example program in Section [Example] illustrates how nag_linsys_complex_gen_norm_rcomm (f04zd) can be used in conjunction with NAG Toolbox function for LU$LU$ factorization of complex matrices nag_lapack_zgetrf (f07ar)).
It is also straightforward to use nag_linsys_complex_gen_norm_rcomm (f04zd) for Hermitian positive definite matrices, using nag_lapack_zpotrf (f07fr) and nag_lapack_zpotrs (f07fs) for factorization and solution.

## Example

```function nag_linsys_complex_gen_norm_rcomm_example
a = [0.7+0.1*i, -0.2+0.0*i,  1.0+0.0*i, 0.0+0.0*i, 0.0+0.0*i, 0.1+0.0*i;
0.3+0.0*i,  0.7+0.0*i,  0.0+0.0*i, 1.0+0.2*i, 0.9+0.0*i, 0.2+0.0*i;
0.0+5.9*i,  0.0+0.0*i,  0.2+0.0*i, 0.7+0.0*i, 0.4+6.1*i, 1.1+0.4*i;
0.0+0.1*i,  0.0+0.1*i, -0.7+0.0*i, 0.2+0.0*i, 0.1+0.0*i, 0.1+0.0*i;
0.0+0.0*i,  4.0+0.0*i,  0.0+0.0*i, 1.0+0.0*i, 9.0+0.0*i, 0.0+0.1*i;
4.5+6.7*i,  0.1+0.4*i,  0.0+3.2*i, 1.2+0.0*i, 0.0+0.0*i, 7.8+0.2*i];
t = int64(2);
m = 6;
n = 6;
x = complex(zeros(n, t));
y = complex(zeros(m, t));
estnrm = 0;
seed = int64(652);
irevcm = int64(0);
work  = complex(zeros(m*t, 1));
rwork = zeros(2*n, 1);
iwork = zeros(2*n+5*t+20, 1, 'int64');

nrma =  norm(a, 1);
fprintf('\nThe norm of a is %6.2f\n', nrma);

% Estimate the norm of a^(-1) without explicitly forming a^(-1)

% Perform an LU factorization so that A=LU where L and U are lower and upper
% triangular.
[a, ipiv, info] = nag_lapack_zgetrf(a);

first = true;

while first || (irevcm ~= 0)
first = false;

[irevcm, x, y, estnrm, work, rwork, iwork, ifail] = ...
nag_linsys_complex_gen_norm_rcomm(irevcm, x, y, estnrm, seed, work, rwork, iwork);

switch irevcm
case 1
% Compute y = inv(a)*x
[y, info] = nag_lapack_zgetrs('n', a, ipiv, x);
case 2
% Compute x = transpose(inv(a))*y
[x, info] = nag_lapack_zgetrs('t', a, ipiv, y);
otherwise
end
end

fprintf('The estimated norm of inverse(a) is: %6.2f\n', estnrm);
fprintf('\nEstimated condition number of a: %6.2f\n', estnrm*nrma);
```
```

The norm of a is  16.11
The estimated norm of inverse(a) is:  24.02

Estimated condition number of a: 387.08

```
```function f04zd_example
a = [0.7+0.1*i, -0.2+0.0*i,  1.0+0.0*i, 0.0+0.0*i, 0.0+0.0*i, 0.1+0.0*i;
0.3+0.0*i,  0.7+0.0*i,  0.0+0.0*i, 1.0+0.2*i, 0.9+0.0*i, 0.2+0.0*i;
0.0+5.9*i,  0.0+0.0*i,  0.2+0.0*i, 0.7+0.0*i, 0.4+6.1*i, 1.1+0.4*i;
0.0+0.1*i,  0.0+0.1*i, -0.7+0.0*i, 0.2+0.0*i, 0.1+0.0*i, 0.1+0.0*i;
0.0+0.0*i,  4.0+0.0*i,  0.0+0.0*i, 1.0+0.0*i, 9.0+0.0*i, 0.0+0.1*i;
4.5+6.7*i,  0.1+0.4*i,  0.0+3.2*i, 1.2+0.0*i, 0.0+0.0*i, 7.8+0.2*i];
t = int64(2);
m = 6;
n = 6;
x = complex(zeros(n, t));
y = complex(zeros(m, t));
estnrm = 0;
seed = int64(652);
irevcm = int64(0);
work  = complex(zeros(m*t, 1));
rwork = zeros(2*n, 1);
iwork = zeros(2*n+5*t+20, 1, 'int64');

nrma =  norm(a, 1);
fprintf('\nThe norm of a is %6.2f\n', nrma);

% Estimate the norm of a^(-1) without explicitly forming a^(-1)

% Perform an LU factorization so that A=LU where L and U are lower and upper
% triangular.
[a, ipiv, info] = f07ar(a);

first = true;

while first || (irevcm ~= 0)
first = false;

[irevcm, x, y, estnrm, work, rwork, iwork, ifail] = ...
f04zd(irevcm, x, y, estnrm, seed, work, rwork, iwork);

switch irevcm
case 1
% Compute y = inv(a)*x
[y, info] = f07as('n', a, ipiv, x);
case 2
% Compute x = transpose(inv(a))*y
[x, info] = f07as('t', a, ipiv, y);
otherwise
end
end

fprintf('The estimated norm of inverse(a) is: %6.2f\n', estnrm);
fprintf('\nEstimated condition number of a: %6.2f\n', estnrm*nrma);
```
```

The norm of a is  16.11
The estimated norm of inverse(a) is:  24.02

Estimated condition number of a: 387.08

```