f04 Chapter Contents
f04 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_linsys_complex_gen_norm_rcomm (f04zdc)

1  Purpose

nag_linsys_complex_gen_norm_rcomm (f04zdc) estimates the $1$-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.

2  Specification

 #include #include
 void nag_linsys_complex_gen_norm_rcomm (Integer *irevcm, Integer m, Integer n, Complex x[], Integer pdx, Complex y[], Integer pdy, double *estnrm, Integer t, Integer seed, Complex work[], double rwork[], Integer iwork[], NagError *fail)

3  Description

nag_linsys_complex_gen_norm_rcomm (f04zdc) computes an estimate (a lower bound) for the $1$-norm
 $A1 = max 1≤j≤n ∑ i=1 m aij$ (1)
of an $m$ by $n$ complex 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 ${A}^{\mathrm{H}}Y$, 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 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 the 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 ${B}^{-\mathrm{H}}Y$ required by nag_linsys_complex_gen_norm_rcomm (f04zdc) 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 in Higham (1988).
Since ${‖A‖}_{\infty }={‖{A}^{\mathrm{H}}‖}_{1}$, nag_linsys_complex_gen_norm_rcomm (f04zdc) can be used to estimate the $\infty$-norm of $A$ by working with ${A}^{\mathrm{H}}$ 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: ${\mathbf{irevcm}}=1$ 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 ${A}^{\mathrm{H}}Y$ and store the result in x, where ${A}^{\mathrm{H}}$ is the complex conjugate transpose; (b) call nag_linsys_complex_gen_norm_rcomm (f04zdc) once again, with all the arguments unchanged.
On intermediate re-entry: irevcm must be unchanged.
On final exit: ${\mathbf{irevcm}}=0$.
2:     mIntegerInput
On entry: the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     nIntegerInput
On initial entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     x[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array x must be at least ${\mathbf{pdx}}×{\mathbf{t}}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=1$, contains the current matrix $X$.
On intermediate re-entry: if ${\mathbf{irevcm}}=2$, must contain ${A}^{\mathrm{H}}Y$.
On final exit: the array is undefined.
5:     pdxIntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
6:     y[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array y must be at least ${\mathbf{pdy}}×{\mathbf{t}}$.
The $\left(i,j\right)$th element of the matrix $Y$ is stored in ${\mathbf{y}}\left[\left(j-1\right)×{\mathbf{pdy}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=2$, contains the current matrix $Y$.
On intermediate re-entry: if ${\mathbf{irevcm}}=1$, 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: ${\mathbf{pdy}}\ge {\mathbf{m}}$.
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 ${‖A‖}_{1}$.
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 ${\mathbf{t}}\ge 2$ then a partly random starting matrix is used in the algorithm.
Suggested value: ${\mathbf{t}}=2$.
Constraint: $1\le {\mathbf{t}}\le {\mathbf{m}}$.
10:   seedIntegerInput
On entry: the seed used for random number generation.
If ${\mathbf{t}}=1$, seed is not used.
Constraint: if ${\mathbf{t}}>1$, ${\mathbf{seed}}\ge 1$.
11:   work[${\mathbf{m}}×{\mathbf{t}}$]ComplexCommunication Array
12:   rwork[$2×{\mathbf{n}}$]doubleCommunication Array
13:   iwork[$2×{\mathbf{n}}+5×{\mathbf{t}}+20$]IntegerCommunication Array
On initial entry: need not be set.
On intermediate re-entry: must not be changed.
14:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{irevcm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irevcm}}=0$, $1$ or $2$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On initial entry, ${\mathbf{irevcm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irevcm}}=0$.
NE_INT_2
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{t}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{t}}\le {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdy}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdy}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{t}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{seed}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{t}}>1$, ${\mathbf{seed}}\ge 1$.
NE_INTERNAL_ERROR
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 ${‖A‖}_{1}$; 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.

8  Parallelism and Performance

Not applicable.

9.1  Timing

For most problems the time taken during calls to nag_linsys_complex_gen_norm_rcomm (f04zdc) will be negligible compared with the time spent evaluating matrix products between calls to nag_linsys_complex_gen_norm_rcomm (f04zdc).
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 $X={A}^{\mathrm{H}}Y$ 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 ${‖{B}^{-1}‖}_{1}$ using a triangular factorization of $B$, nag_linsys_complex_gen_norm_rcomm (f04zdc) 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 $t\ge 2$, 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.

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 {
f04zdc(&irevcm,m,n,x,pdx,y,pdy,&estnrm,t,seed,work,rwork,iwork,&fail);
if (irevcm == 1){
.. Code to compute y = A^(-1) x ..
}
else if  (irevcm == 2){
.. Code to compute x = A^(-H) y ..
}
} (while irevcm != 0)
```
To compute ${A}^{-1}X$ or ${A}^{-\mathrm{H}}Y$, solve the equation $AY=X$ or ${A}^{\mathrm{H}}X=Y$ storing the result in y or x respectively. The code will vary, depending on the type of the matrix $A$, and the NAG function used to factorize $A$.
The example program in Section 10 illustrates how nag_linsys_complex_gen_norm_rcomm (f04zdc) can be used in conjunction with NAG C Library function for $LU$ factorization of complex matrices nag_zgetrf (f07arc)).
It is also straightforward to use nag_linsys_complex_gen_norm_rcomm (f04zdc) for Hermitian positive definite matrices, using nag_zge_copy (f16tfc)nag_zpotrf (f07frc) and nag_zpotrs (f07fsc) for factorization and solution.
For upper or lower triangular square matrices, no factorization function is needed: $Y={A}^{-1}X$ and $X={A}^{-\mathrm{H}}Y$ may be computed by calls to nag_ztrsv (f16sjc) (or nag_ztbsv (f16skc) if the matrix is banded, or nag_ztpsv (f16slc) if the matrix is stored in packed form).

10  Example

This example estimates the condition number ${‖A‖}_{1}{‖{A}^{-1}‖}_{1}$ of the matrix $A$ given by
 $A = 0.7+0.1i -0.2+0.0i 1.0+0.0i 0.0+0.0i 0.0+0.0i 0.1+0.0i 0.3+0.0i 0.7+0.0i 0.0+0.0i 1.0+0.2i 0.9+0.0i 0.2+0.0i 0.0+5.9i 0.0+0.0i 0.2+0.0i 0.7+0.0i 0.4+6.1i 1.1+0.4i 0.0+0.1i 0.0+0.1i -0.7+0.0i 0.2+0.0i 0.1+0.0i 0.1+0.0i 0.0+0.0i 4.0+0.0i 0.0+0.0i 1.0+0.0i 9.0+0.0i 0.0+0.1i 4.5+6.7i 0.1+0.4i 0.0+3.2i 1.2+0.0i 0.0+0.0i 7.8+0.2i .$

10.1  Program Text

Program Text (f04zdce.c)

10.2  Program Data

Program Data (f04zdce.d)

10.3  Program Results

Program Results (f04zdce.r)