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NAG Toolbox: nag_lapack_dopgtr (f08gf)

Purpose

nag_lapack_dopgtr (f08gf) generates the real orthogonal matrix QQ, which was determined by nag_lapack_dsptrd (f08ge) when reducing a symmetric matrix to tridiagonal form.

Syntax

[q, info] = f08gf(uplo, n, ap, tau)
[q, info] = nag_lapack_dopgtr(uplo, n, ap, tau)

Description

nag_lapack_dopgtr (f08gf) is intended to be used after a call to nag_lapack_dsptrd (f08ge), which reduces a real symmetric matrix AA to symmetric tridiagonal form TT by an orthogonal similarity transformation: A = QTQTA=QTQT. nag_lapack_dsptrd (f08ge) represents the orthogonal matrix QQ as a product of n1n-1 elementary reflectors.
This function may be used to generate QQ explicitly as a square matrix.

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)
This must be the same parameter uplo as supplied to nag_lapack_dsptrd (f08ge).
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix QQ.
Constraint: n0n0.
3:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).
4:     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 elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

ldq work

Output Parameters

1:     q(ldq, : :) – double array
The first dimension of the array q will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldqmax (1,n)ldqmax(1,n).
The nn by nn orthogonal matrix QQ.
2:     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: uplo, 2: n, 3: ap, 4: tau, 5: q, 6: ldq, 7: work, 8: 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 matrix QQ differs from an exactly orthogonal matrix by a matrix EE such that
E2 = O(ε) ,
E2 = O(ε) ,
where εε is the machine precision.

Further Comments

The total number of floating point operations is approximately (4/3)n343n3.
The complex analogue of this function is nag_lapack_zupgtr (f08gt).

Example

function nag_lapack_dopgtr_example
uplo = 'L';
n = int64(4);
ap = [2.07;
     -5.825753170191817;
     0.4331793442217867;
     -0.1186086299654892;
     1.474093708197552;
     2.624045178795586;
     0.8062881532775791;
     -0.6491595075457843;
     0.9162727563219193;
     -1.694934200651768];
tau = [1.664291789738249;
     1.212047324162142;
     0];
[q, info] = nag_lapack_dopgtr(uplo, n, ap, tau)
 

q =

    1.0000         0         0         0
         0   -0.6643   -0.0400    0.7464
         0   -0.7209   -0.2294   -0.6539
         0    0.1974   -0.9725    0.1235


info =

                    0


function f08gf_example
uplo = 'L';
n = int64(4);
ap = [2.07;
     -5.825753170191817;
     0.4331793442217867;
     -0.1186086299654892;
     1.474093708197552;
     2.624045178795586;
     0.8062881532775791;
     -0.6491595075457843;
     0.9162727563219193;
     -1.694934200651768];
tau = [1.664291789738249;
     1.212047324162142;
     0];
[q, info] = f08gf(uplo, n, ap, tau)
 

q =

    1.0000         0         0         0
         0   -0.6643   -0.0400    0.7464
         0   -0.7209   -0.2294   -0.6539
         0    0.1974   -0.9725    0.1235


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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