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NAG Toolbox

NAG Toolbox: nag_lapack_zhptrd (f08gs)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zhptrd (f08gs) reduces a complex Hermitian matrix to tridiagonal form, using packed storage.

Syntax

[ap, d, e, tau, info] = f08gs(uplo, n, ap)
[ap, d, e, tau, info] = nag_lapack_zhptrd(uplo, n, ap)

Description

nag_lapack_zhptrd (f08gs) reduces a complex Hermitian matrix A, held in packed storage, to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH.
The matrix Q is not formed explicitly but is represented as a product of n-1 elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Further Comments).

References

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A is stored.
uplo='U'
The upper triangular part of A is stored.
uplo='L'
The lower triangular part of A is stored.
Constraint: uplo='U' or 'L'.
2:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
3:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The upper or lower triangle of the n by n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
ap stores the tridiagonal matrix T and details of the unitary matrix Q.
2:     dn – double array
The diagonal elements of the tridiagonal matrix T.
3:     en-1 – double array
The off-diagonal elements of the tridiagonal matrix T.
4:     taun-1 – complex array
Further details of the unitary matrix Q.
5:     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: uplo, 2: n, 3: ap, 4: d, 5: e, 6: tau, 7: info.

Accuracy

The computed tridiagonal matrix T is exactly similar to a nearby matrix A+E, where
E2 cn ε A2 ,  
cn is a modestly increasing function of n, and ε is the machine precision.
The elements of T themselves may be sensitive to small perturbations in A or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

Further Comments

The total number of real floating-point operations is approximately 163 n3 .
To form the unitary matrix Q nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupgtr (f08gt):
[q, info] = f08gt(uplo, n, ap, tau);
To apply Q to an n by p complex matrix C nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupmtr (f08gu). For example,
[ap, c, info] = f08gu('Left', uplo, 'No Transpose', ap, tau, c);
forms the matrix product QC.
The real analogue of this function is nag_lapack_dsptrd (f08ge).

Example

This example reduces the matrix A to tridiagonal form, 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.
function f08gs_example


fprintf('f08gs 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);

% Note: absolute values for e are displayed because the signs may change
%       with changes in sign of columns of Q.
fprintf('Diagonal and off-diagonal elements of tridiagonal form\n\n');
fprintf('    i         d           e\n');

for j = 1:n-1
  fprintf('%5d%12.5f%12.5f\n', j, d(j), abs(e(j)));
end
fprintf('%5d%12.5f\n', n, d(n));


f08gs example results

Diagonal and off-diagonal elements of tridiagonal form

    i         d           e
    1    -2.28000     4.33846
    2    -0.12846     2.02259
    3    -0.16659     1.80232
    4    -1.92495

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