nag_linsys_real_gen_norm_rcomm (f04ydc) (PDF version)
f04 Chapter Contents
f04 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_linsys_real_gen_norm_rcomm (f04ydc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_linsys_real_gen_norm_rcomm (f04ydc) 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.

2  Specification

#include <nag.h>
#include <nagf04.h>
void  nag_linsys_real_gen_norm_rcomm (Integer *irevcm, Integer m, Integer n, double x[], Integer pdx, double y[], Integer pdy, double *estnrm, Integer t, Integer seed, double work[], Integer iwork[], NagError *fail)

3  Description

nag_linsys_real_gen_norm_rcomm (f04ydc) computes an estimate (a lower bound) for the 1-norm
A1 = max 1jn i=1 m aij (1)
of an m by n real matrix A=aij. The function regards the matrix A as being defined by a user-supplied ‘Black Box’ which, given an n×t matrix X (with tn) or an m×t matrix Y, can return AX or ATY. 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-11 for a square matrix, B, and hence the condition number κ1B=B1B-11, without forming B-1 explicitly (A=B-1 above).
If, for example, an LU factorization of B is available, the matrix products B-1X and B-TY required by nag_linsys_real_gen_norm_rcomm (f04ydc) 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-1B and ABC, without forming the products explicitly. Further applications are described by Higham (1988).
Since A=AT1, nag_linsys_real_gen_norm_rcomm (f04ydc) can be used to estimate the -norm of A by working with AT instead of A.
The algorithm used is described in Higham and Tisseur (2000).

4  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

5  Arguments

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.
1:     irevcmInteger *Input/Output
On initial entry: must be set to 0.
On intermediate exit: 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 irevcm=1, evaluate AX and store the result in y
if irevcm=2, evaluate ATY and store the result in x,
(b) call nag_linsys_real_gen_norm_rcomm (f04ydc) once again, with all the other arguments unchanged.
On intermediate re-entry: irevcm must be unchanged.
On final exit: irevcm=0.
2:     mIntegerInput
On entry: the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     x[dim]doubleInput/Output
Note: the dimension, dim, of the array x must be at least pdx×t.
The i,jth element of the matrix X is stored in x[j-1×pdx+i-1].
On initial entry: need not be set.
On intermediate exit: if irevcm=1, contains the current matrix X.
On intermediate re-entry: if irevcm=2, must contain ATY.
On final exit: the array is undefined.
5:     pdxIntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraint: pdxn.
6:     y[dim]doubleInput/Output
Note: the dimension, dim, of the array y must be at least pdy×t.
The i,jth element of the matrix Y is stored in y[j-1×pdy+i-1].
On initial entry: need not be set.
On intermediate exit: if irevcm=2, contains the current matrix Y.
On intermediate re-entry: if irevcm=1, must contain AX.
On final exit: the array is undefined.
7:     pdyIntegerInput
On entry: the stride separating matrix row elements in the array y.
Constraint: pdym.
8:     estnrmdouble *Input/Output
On initial entry: need not be set.
On intermediate re-entry: must not be changed.
On final exit: an estimate (a lower bound) for A1.
9:     tIntegerInput
On entry: the number of columns t of the matrices X and Y. This is an 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 t2 then a partly random starting matrix is used in the algorithm.
Suggested value: t=2.
Constraint: 1tm.
10:   seedIntegerInput
On entry: the seed used for random number generation.
If t=1, seed is not used.
Constraint: if t>1, seed1.
11:   work[m×t]doubleCommunication Array
12:   iwork[2×n+5×t+20]IntegerCommunication Array
On initial entry: need not be set.
On intermediate re-entry: must not be changed.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, irevcm=value.
Constraint: irevcm=0, 1 or 2.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On initial entry, irevcm=value.
Constraint: irevcm=0.
On entry, m=value and t=value.
Constraint: 1tm.
On entry, pdx=value and n=value.
Constraint: pdxn.
On entry, pdy=value and m=value.
Constraint: pdym.
On entry, t=value and seed=value.
Constraint: if t>1, seed1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  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 A1; often the estimate has many correct figures. However, matrices exist for which the estimate is smaller than A1 by an arbitrary factor; such matrices are very unlikely to arise in practice. See Higham and Tisseur (2000) for further details.

8  Parallelism and Performance

Not applicable.

9  Further Comments

9.1  Timing

For most problems the time taken during calls to nag_linsys_real_gen_norm_rcomm (f04ydc) will be negligible compared with the time spent evaluating matrix products between calls to nag_linsys_real_gen_norm_rcomm (f04ydc).
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=ATY will be required. The number of iterations is independent of the choice of t.

9.2  Overflow

It is your responsibility to guard against potential overflows during evaluation of the matrix products. In particular, when estimating B-11 using a triangular factorization of B, nag_linsys_real_gen_norm_rcomm (f04ydc) should not be called if one of the factors is exactly singular – otherwise division by zero may occur in the substitutions.

9.3  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 t2, 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.

9.4  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:
do {
   if (irevcm == 1){
     .. Code to compute y = A^(-1) x ..
   else if  (irevcm == 2){
     .. Code to compute x = A^(-T) y ..
} (while irevcm != 0)
To compute A-1X or A-TY, solve the equation AY=X or ATX=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 f01f03 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 Section 10 illustrates how nag_linsys_real_gen_norm_rcomm (f04ydc) can be used in conjunction with NAG C Library functions for LU factorization of a real matrix nag_dgetrf (f07adc).
It is straightforward to use nag_linsys_real_gen_norm_rcomm (f04ydc) for the following other types of matrix, using the named functions for factorization and solution:
For upper or lower triangular matrices, no factorization function is needed: Y=A-1X and X=A-TY may be computed by calls to nag_dtrsv (f16pjc) (or nag_dtbsv (f16pkc) if the matrix is banded, or nag_dtpsv (f16plc) if the matrix is stored in packed form).

10  Example

This example estimates the condition number A1A-11 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 .

10.1  Program Text

Program Text (f04ydce.c)

10.2  Program Data

Program Data (f04ydce.d)

10.3  Program Results

Program Results (f04ydce.r)

nag_linsys_real_gen_norm_rcomm (f04ydc) (PDF version)
f04 Chapter Contents
f04 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014