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NAG Toolbox: nag_lapack_zhetrd (f08fs)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zhetrd (f08fs) reduces a complex Hermitian matrix to tridiagonal form.

Syntax

[a, d, e, tau, info] = f08fs(uplo, a, 'n', n)
[a, d, e, tau, info] = nag_lapack_zhetrd(uplo, a, 'n', n)

Description

nag_lapack_zhetrd (f08fs) reduces a complex Hermitian matrix A 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:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n Hermitian matrix A.
  • If uplo='U', the upper triangular part of a must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of a must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
a stores the tridiagonal matrix T and details of the unitary matrix Q as specified by uplo.
2:     d: – double array
The dimension of the array d will be max1,n
The diagonal elements of the tridiagonal matrix T.
3:     e: – double array
The dimension of the array e will be max1,n-1
The off-diagonal elements of the tridiagonal matrix T.
4:     tau: – complex array
The dimension of the array tau will be max1,n-1
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: a, 4: lda, 5: d, 6: e, 7: tau, 8: work, 9: lwork, 10: 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 tridiagonal matrix T is exactly similar to a nearby matrix A+E, where
E2 c n 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_zhetrd (f08fs) may be followed by a call to nag_lapack_zungtr (f08ft):
[a, info] = f08ft(uplo, a, tau);
To apply Q to an n by p complex matrix C nag_lapack_zhetrd (f08fs) may be followed by a call to nag_lapack_zunmtr (f08fu). For example,
[c, info] = f08fu('Left', uplo, 'No Transpose', a, tau, c);
forms the matrix product QC.
The real analogue of this function is nag_lapack_dsytrd (f08fe).

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 .  
function f08fs_example


fprintf('f08fs example results\n\n');

% A --> QTQ^H, for tridiagonal T
uplo = 'L';
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

[QT, d, e, tau, info] = f08fs(uplo, a);

disp('Main diagonal of T:');
disp (d');
disp('Off diagonal of T (absolute values):');
disp (abs(e)');


f08fs example results

Main diagonal of T:
   -2.2800   -0.1285   -0.1666   -1.9249

Off diagonal of T (absolute values):
    4.3385    2.0226    1.8023


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