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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zupmtr (f08gu)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zupmtr (f08gu) multiplies an arbitrary complex matrix C by the complex unitary matrix Q which was determined by nag_lapack_zhptrd (f08gs) when reducing a complex Hermitian matrix to tridiagonal form.

Syntax

[ap, c, info] = f08gu(side, uplo, trans, ap, tau, c, 'm', m, 'n', n)
[ap, c, info] = nag_lapack_zupmtr(side, uplo, trans, ap, tau, c, 'm', m, 'n', n)

Description

nag_lapack_zupmtr (f08gu) is intended to be used after a call to nag_lapack_zhptrd (f08gs), which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. nag_lapack_zhptrd (f08gs) represents the unitary matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QHC , CQ ​ or ​ CQH ,  
overwriting the result on C (which may be any complex rectangular matrix).
A common application of this function is to transform a matrix Z of eigenvectors of T to the matrix QZ of eigenvectors of A.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     side – string (length ≥ 1)
Indicates how Q or QH is to be applied to C.
side='L'
Q or QH is applied to C from the left.
side='R'
Q or QH is applied to C from the right.
Constraint: side='L' or 'R'.
2:     uplo – string (length ≥ 1)
This must be the same argument uplo as supplied to nag_lapack_zhptrd (f08gs).
Constraint: uplo='U' or 'L'.
3:     trans – string (length ≥ 1)
Indicates whether Q or QH is to be applied to C.
trans='N'
Q is applied to C.
trans='C'
QH is applied to C.
Constraint: trans='N' or 'C'.
4:     ap: – complex array
The dimension of the array ap must be at least max1, m × m+1 / 2  if side='L' and at least max1, n × n+1 / 2  if side='R'
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zhptrd (f08gs).
5:     tau: – complex array
The dimension of the array tau must be at least max1,m-1 if side='L' and at least max1,n-1 if side='R'
Further details of the elementary reflectors, as returned by nag_lapack_zhptrd (f08gs).
6:     cldc: – complex array
The first dimension of the array c must be at least max1,m.
The second dimension of the array c must be at least max1,n.
The m by n matrix C.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array c.
m, the number of rows of the matrix C; m is also the order of Q if side='L'.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array c.
n, the number of columns of the matrix C; n is also the order of Q if side='R'.
Constraint: n0.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1, m × m+1 / 2  if side='L' and at least max1, n × n+1 / 2  if side='R'
Is used as internal workspace prior to being restored and hence is unchanged.
2:     cldc: – complex array
The first dimension of the array c will be max1,m.
The second dimension of the array c will be max1,n.
c stores QC or QHC or CQ or CQH as specified by side and trans.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: side, 2: uplo, 3: trans, 4: m, 5: n, 6: ap, 7: tau, 8: c, 9: ldc, 10: work, 11: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed result differs from the exact result by a matrix E such that
E2 = Oε C2 ,  
where ε is the machine precision.

Further Comments

The total number of real floating-point operations is approximately 8m2n if side='L' and 8mn2 if side='R'.
The real analogue of this function is nag_lapack_dopmtr (f08gg).

Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix A, where
A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,  
using packed storage. Here A is Hermitian and must first be reduced to tridiagonal form T by nag_lapack_zhptrd (f08gs). The program then calls nag_lapack_dstebz (f08jj) to compute the requested eigenvalues and nag_lapack_zstein (f08jx) to compute the associated eigenvectors of T. Finally nag_lapack_zupmtr (f08gu) is called to transform the eigenvectors to those of A.
function f08gu_example


fprintf('f08gu example results\n\n');

% Hermitian matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [-2.28 + 0i;   1.78 + 2.03i;   2.26 - 0.10i;  -0.12 - 2.53i;
                   -1.12 + 0i;      0.01 - 0.43i;  -1.07 - 0.86i;
                                   -0.37 + 0i;      2.31 + 0.92i;
                                                   -0.73 + 0i];

% Reduce to tridiagonal form
[apf, d, e, tau, info] = f08gs( ...
                                uplo, n, ap);

% Calculate two smallest eigenvalues
range = 'Indices';
order = 'Block';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, nsplit, w, iblock, isplit, info] = ...
  f08jj( ...
         range, order, vl, vu, il, iu, abstol, d, e);

% Corresponding eigenvectors of T
[tz, ifailv, info] = f08jx( ...
                            d, e, m, w, iblock, isplit);

% Transform to eigenvalues of A (by premultiplying by Q)
side = 'Left';
trans = 'No transpose';
[~, z, info] = f08gu( ...
                      side, uplo, trans, apf, tau, tz);

% Normalize vectors, largest element is real and positive.
for i = 1:m
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp(' Selected eigenvalues of A:');
disp(w(1:m));
disp(' Corresponding eigenvectors:');
disp(z);


f08gu example results

 Selected eigenvalues of A:
   -6.0002
   -3.0030

 Corresponding eigenvectors:
   0.7299 + 0.0000i  -0.2120 + 0.1497i
  -0.1663 - 0.2061i   0.7307 + 0.0000i
  -0.4165 - 0.1417i  -0.3291 + 0.0479i
   0.1743 + 0.4162i   0.5200 + 0.1329i


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Chapter Contents
Chapter Introduction
NAG Toolbox

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