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NAG Toolbox: nag_lapack_dgehrd (f08ne)

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 AA to upper Hessenberg form HH by an orthogonal similarity transformation: A = QHQTA=QHQT.
The matrix QQ 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 QQ in this representation (see Section [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  
A11 A12 A13 A22 A23 A33
where A11A11 and A33A33 are upper triangular. If so, only the central diagonal block A22A22, in rows and columns iloilo to ihiihi, needs to be reduced to Hessenberg form (the blocks A12A12 and A23A23 will also be affected by the reduction). Therefore the values of iloilo and ihiihi 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 iloilo must be set to 11 and ihiihi to nn.

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 AA 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 11 and ihi to n.
Constraints:
  • if n > 0n>0, 1 ilo ihi n 1 ilo ihi n ;
  • if n = 0n=0, ilo = 1ilo=1 and ihi = 0ihi=0.
3:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn general matrix AA.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work lwork

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
a stores the upper Hessenberg matrix HH and details of the orthogonal matrix QQ.
2:     tau( : :) – double array
Note: the dimension of the array tau must be at least max (1,n1)max(1,n-1).
Further details of the orthogonal matrix QQ.
3:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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 HH is exactly similar to a nearby matrix (A + E)(A+E), where
E2 c (n) ε A2 ,
E2 c (n) ε A2 ,
c(n)c(n) is a modestly increasing function of nn, and εε is the machine precision.
The elements of HH themselves may be sensitive to small perturbations in AA 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 (2/3)q2(2q + 3n)23q2(2q+3n), where q = ihiiloq=ihi-ilo; if ilo = 1ilo=1 and ihi = nihi=n, the number is approximately (10/3)n3103n3.
To form the orthogonal matrix QQ nag_lapack_dgehrd (f08ne) may be followed by a call to nag_lapack_dorghr (f08nf):
[a, info] = f08nf(ilo, ihi, a, tau);
To apply QQ to an mm by nn real matrix CC 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 QCQC.
The complex analogue of this function is nag_lapack_zgehrd (f08ns).

Example

function nag_lapack_dgehrd_example
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];
[aOut, tau, info] = nag_lapack_dgehrd(ilo, ihi, a)
 

aOut =

    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


tau =

    1.1751
    1.9460
         0


info =

                    0


function f08ne_example
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];
[aOut, tau, info] = f08ne(ilo, ihi, a)
 

aOut =

    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


tau =

    1.1751
    1.9460
         0


info =

                    0



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