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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_real_gen_norm_rcomm (f04yd)

## Purpose

nag_linsys_real_gen_norm_rcomm (f04yd) estimates the $1$-norm of a real 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, iwork, ifail] = f04yd(irevcm, x, y, estnrm, seed, work, iwork, 'm', m, 'n', n, 't', t)
[irevcm, x, y, estnrm, work, iwork, ifail] = nag_linsys_real_gen_norm_rcomm(irevcm, x, y, estnrm, seed, work, iwork, 'm', m, 'n', n, 't', t)

## Description

nag_linsys_real_gen_norm_rcomm (f04yd) computes an estimate (a lower bound) for the $1$-norm
 $A1 = max 1≤j≤n ∑ i=1 m aij$ (1)
of an $m$ by $n$ real matrix $A=\left({a}_{ij}\right)$. The function regards the matrix $A$ as being defined by a user-supplied ‘Black Box’ which, given an $n×t$ matrix $X$ (with $t\ll n$) or an $m×t$ matrix $Y$, can return $AX$ or ${A}^{\mathrm{T}}Y$. 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 $A$ are known explicitly; it is then more efficient to compute the $1$-norm directly from formula (1) above.
The main use of the function is for estimating ${‖{B}^{-1}‖}_{1}$ for a square matrix, $B$, and hence the condition number ${\kappa }_{1}\left(B\right)={‖B‖}_{1}{‖{B}^{-1}‖}_{1}$, without forming ${B}^{-1}$ explicitly ($A={B}^{-1}$ above).
If, for example, an $LU$ factorization of $B$ is available, the matrix products ${B}^{-1}X$ and ${B}^{-\mathrm{T}}Y$ required by nag_linsys_real_gen_norm_rcomm (f04yd) may be computed by back- and forward-substitutions, without computing ${B}^{-1}$.
The function 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 by Higham (1988).
Since ${‖A‖}_{\infty }={‖{A}^{\mathrm{T}}‖}_{1}$, nag_linsys_real_gen_norm_rcomm (f04yd) can be used to estimate the $\infty$-norm of $A$ by working with ${A}^{\mathrm{T}}$ instead of $A$.
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 $1$-norm estimation, with an application to $1$-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 argument irevcm. Between intermediate exits and re-entries, all arguments other than x and y must remain unchanged.

### Compulsory Input Parameters

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

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{irevcm}$int64int32nag_int scalar
On intermediate exit: ${\mathbf{irevcm}}=1$ or $2$, and x contains the $n×t$ matrix $X$ and y contains the $m×t$ matrix $Y$. The calling program must
 (a) if ${\mathbf{irevcm}}=1$, evaluate $AX$ and store the result in y or if ${\mathbf{irevcm}}=2$, evaluate ${A}^{\mathrm{T}}Y$ and store the result in x, (b) call nag_linsys_real_gen_norm_rcomm (f04yd) once again, with all the other arguments unchanged.
On final exit: ${\mathbf{irevcm}}=0$.
2:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x will be ${\mathbf{n}}$.
The second dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$.
On intermediate exit: if ${\mathbf{irevcm}}=1$, contains the current matrix $X$.
On final exit: the array is undefined.
3:     $\mathrm{y}\left(\mathit{ldy},:\right)$ – double array
The first dimension of the array y will be ${\mathbf{m}}$.
The second dimension of the array y will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{t}}\right)$.
On intermediate exit: if ${\mathbf{irevcm}}=2$, contains the current matrix $Y$.
On final exit: the array is undefined.
4:     $\mathrm{estnrm}$ – double scalar
On final exit: an estimate (a lower bound) for ${‖A‖}_{1}$.
5:     $\mathrm{work}\left({\mathbf{m}}×{\mathbf{t}}\right)$ – double array
6:     $\mathrm{iwork}\left(2×{\mathbf{n}}+5×{\mathbf{t}}+20\right)$int64int32nag_int array
7:     $\mathrm{ifail}$int64int32nag_int scalar
On final exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{irevcm}}=0$, $1$ or $2$.
On initial entry, ${\mathbf{irevcm}}=_$.
Constraint: ${\mathbf{irevcm}}=0$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{m}}\ge 0$.
${\mathbf{ifail}}=-3$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-5$
Constraint: $\mathit{ldx}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-7$
Constraint: $\mathit{ldy}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=-9$
Constraint: $1\le {\mathbf{t}}\le {\mathbf{m}}$.
${\mathbf{ifail}}=-10$
Constraint: if ${\mathbf{t}}>1$, ${\mathbf{seed}}\ge 1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

In extensive tests on random matrices of size up to $m=n=450$ the estimate estnrm has been found always to be within a factor two of ${‖A‖}_{1}$; often the estimate has many correct figures. However, matrices exist for which the estimate is smaller than ${‖A‖}_{1}$ 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_real_gen_norm_rcomm (f04yd) will be negligible compared with the time spent evaluating matrix products between calls to nag_linsys_real_gen_norm_rcomm (f04yd).
The number of matrix products required depends on the matrix $A$. At most six products of the form $Y=AX$ and five products of the form $X={A}^{\mathrm{T}}Y$ will be required. The number of iterations is independent of the choice of $t$.

### Overflow

It is your responsibility to guard against potential overflows during evaluation of the matrix products. In particular, when estimating ${‖{B}^{-1}‖}_{1}$ using a triangular factorization of $B$, nag_linsys_real_gen_norm_rcomm (f04yd) 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 argument $t$ controls the accuracy and reliability of the estimate. For $t=1$, the algorithm behaves similarly to the LAPACK estimator xLACON. Increasing $t$ typically improves the estimate, without increasing the number of iterations required.
For $t\ge 2$, random matrices are used in the algorithm, so for repeatable results the same value of seed should be used each time.
A value of $t=2$ is recommended for new users.

### Use in Conjunction with NAG Library Routines

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)
icase = 0;
[icase, x, estnrm, work, iwork, ifail] = f04yd(icase, x, estnrm, work, iwork);
while (icase ~= 0)
if (icase == 1)
...  code to compute inv(A)*x ...
else
...  code to compute inv(transpose(A))*x ...
end
[icase, x, estnrm, work, iwork, ifail] = f04yd(icase, x, estnrm, work, iwork);
end
end```
To compute ${A}^{-1}X$ or ${A}^{-\mathrm{T}}Y$, solve the equation $AY=X$ or ${A}^{\mathrm{T}}X=Y$, storing the result in y or x respectively. The code will vary, depending on the type of the matrix $A$, and the NAG function used to factorize $A$.
The factorization will normally have been performed by a suitable function from Chapters F01, F03 or F07. Note also that many of the ‘Black Box’ functions in Chapter F04 for solving systems of equations also return a factorization of the matrix. The example program in Example illustrates how nag_linsys_real_gen_norm_rcomm (f04yd) can be used in conjunction with NAG Toolbox functions for $LU$ factorization of a real matrix nag_lapack_dgetrf (f07ad).
It is straightforward to use nag_linsys_real_gen_norm_rcomm (f04yd) for the following other types of matrix, using the named functions for factorization and solution:

## Example

For this function two examples are provided. There is a single example program for nag_linsys_real_gen_norm_rcomm (f04yd), with a main program and the code to solve the two example problems is given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
This example estimates the condition number ${‖A‖}_{1}{‖{A}^{-1}‖}_{1}$ of the matrix $A$ given by
 $A= 0.7 -0.2 1.0 0.0 2.0 0.1 0.3 0.7 0.0 1.0 0.9 0.2 0.0 0.0 0.2 0.7 0.0 -1.1 0.0 3.4 -0.7 0.2 0.1 0.1 0.0 -4.0 0.0 1.0 9.0 0.0 0.4 1.2 4.3 0.0 6.2 5.9 .$
Example 2 (EX2)
This example estimates the condition number of the sparse matrix $A$ (stored in symmetric coordinate storage format) given by
 $A = 0.0 0.0 0.0 1.0 0.0 3.0 1.0 0.0 0.0 0.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 4.0 0.0 5.0 0.0 1.0 2.0 0.0 0.0 .$
```function f04yd_example

fprintf('f04yd example results\n\n');

% Example 1: Compute the condition number of a dense matrix
fprintf('\nExample 1\n');

a = [0.7,  -0.2,   1.0,   0.0,   2.0,   0.1;
0.3,   0.7,   0.0,   1.0,   0.9,   0.2;
0.0,   0.0,   0.2,   0.7,   0.0,  -1.1;
0.0,   3.4,  -0.7,   0.2,   0.1,   0.1;
0.0,  -4.0,   0.0,   1.0,   9.0,   0.0;
0.4,   1.2,   4.3,   0.0,   6.2,   5.9];
t = int64(2);
m = 6;
n = 6;
x = zeros(n, t);
y = zeros(m, t);
estnrm = 0;
seed = int64(354);
irevcm = int64(0);
work = zeros(n*t, 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.

first = true;

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

[irevcm, x, y, estnrm, work, iwork, ifail] = ...
f04yd( ...
irevcm, x, y, estnrm, seed, work, iwork);

switch irevcm
case 1
% Compute y = inv(a)*x
[y, info] = f07ae('n', a, ipiv, x);
case 2
% Compute x = transpose(inv(a))*y
[x, info] = f07ae('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);

% Example 2: Compute the condition number of a sparse matrix
% (stored in symmetric coordinate storage format)
fprintf('\nExample 2\n');

t = int64(2);
n = int64(5);
nz = int64(10);
a = zeros(4*nz, 1);
icn = zeros(4*nz, 1, 'int64');
irn = zeros(4*nz, 1, 'int64');
a(1:nz)   = [3; 2; 1; 2; 1; 4; 2; 1; 2; 5];
irn(1:nz) = [2; 4; 2; 3; 5; 4; 5; 1; 3; 4];
icn(1:nz) = [1; 1; 2; 2; 2; 3; 3; 4; 4; 5];

x = zeros(n, t);
y = zeros(n, t);
estnrm = 0;
seed = int64(412);
irevcm = int64(0);
work = zeros(n*t, 1);
iwork = zeros(2*n+5*t+20, 1, 'int64');

% Compute 1-norm of a
nrma =  0;
for i = 1:n
asum = 0;
for j = 1:nz
if (icn(j)==i)
asum = asum + abs(a(j));
end
end
nrma = max(nrma,asum);
end

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

% Perform an LU factorization so that A=LU where L and U are lower and upper
% triangular.
abort = [true; true; false; false];
[a, irn, icn, ikeep, w, idisp, ifail] = ...
f01br(n, nz, a, irn, icn, abort);

% Compute an estimate of the 1-norm of inv(a)

first = true;

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

[irevcm, x, y, estnrm, work, iwork, ifail] = ...
f04yd( ...
irevcm, x, y, estnrm, seed, work, iwork);

switch irevcm
case 1
% Compute y = inv(a)*x
for i=1:2
[y(:, i), resid] = f04ax( ...
a, icn, ikeep, x(:, i), irevcm, idisp);
end
case 2
% Compute x = transpose(inv(a))*y
for i=1:2
[x(:, i), resid] = f04ax( ...
a, icn, ikeep, y(:, i), irevcm, idisp);
end
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);

```
```f04yd example results

Example 1

The norm of a is  18.20
The estimated norm of inverse(a) is:   2.97

Estimated condition number of a:  54.14

Example 2

The norm of a is   6.00
The estimated norm of inverse(a) is:   3.37

Estimated condition number of a:  20.20
```