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NAG Toolbox: nag_lapack_dgehrd (f08ne)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgehrd (f08ne) reduces a real general matrix to Hessenberg form.

Syntax

[a, tau, info] = f08ne(ilo, ihi, a, 'n', n)
[a, tau, info] = nag_lapack_dgehrd(ilo, ihi, a, 'n', n)

Description

nag_lapack_dgehrd (f08ne) reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT.
The matrix Q is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Further Comments).
The function can take advantage of a previous call to nag_lapack_dgebal (f08nh), which may produce a matrix with the structure:
A11 A12 A13 A22 A23 A33  
where A11 and A33 are upper triangular. If so, only the central diagonal block A22, in rows and columns ilo to ihi, needs to be reduced to Hessenberg form (the blocks A12 and A23 will also be affected by the reduction). Therefore the values of ilo and ihi determined by nag_lapack_dgebal (f08nh) can be supplied to the function directly. If nag_lapack_dgebal (f08nh) has not previously been called however, then ilo must be set to 1 and ihi to n.

References

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

Parameters

Compulsory Input Parameters

1:     ilo int64int32nag_int scalar
2:     ihi int64int32nag_int scalar
If A has been output by nag_lapack_dgebal (f08nh), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.
Constraints:
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
3:     alda: – double 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 general matrix A.

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: – double 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 upper Hessenberg matrix H and details of the orthogonal matrix Q.
2:     tau: – double array
The dimension of the array tau will be max1,n-1
Further details of the orthogonal matrix Q.
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: n, 2: ilo, 3: ihi, 4: a, 5: lda, 6: tau, 7: work, 8: lwork, 9: 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 Hessenberg matrix H 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 H 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, eigenvectors or Schur factorization.

Further Comments

The total number of floating-point operations is approximately 23q22q+3n, where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 103n3.
To form the orthogonal matrix Q nag_lapack_dgehrd (f08ne) may be followed by a call to nag_lapack_dorghr (f08nf):
[a, info] = f08nf(ilo, ihi, a, tau);
To apply Q to an m by n real matrix C nag_lapack_dgehrd (f08ne) may be followed by a call to nag_lapack_dormhr (f08ng). For example,
[c, info] = f08ng('Left', 'No Transpose', ilo, ihi, a, tau, c);
forms the matrix product QC.
The complex analogue of this function is nag_lapack_zgehrd (f08ns).

Example

This example computes the upper Hessenberg form of the matrix A, where
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .  
function f08ne_example


fprintf('f08ne example results\n\n');

ilo = int64(1);
ihi = int64(4);
a = [ 0.35,  0.45, -0.14, -0.17;
      0.09,  0.07, -0.54,  0.35;
     -0.44, -0.33, -0.03,  0.17;
      0.25, -0.32, -0.13,  0.11];

% Reduce A to upper Hessenberg Form
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

disp('Upper Hessenberg Form H');
disp(H);


f08ne example results

Upper Hessenberg Form H
    0.3500   -0.1160   -0.3886   -0.2942
   -0.5140    0.1225    0.1004    0.1126
   -0.7285    0.6443   -0.1357   -0.0977
    0.4139   -0.1665    0.4262    0.1632


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