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NAG Toolbox: nag_lapack_zupgtr (f08gt)

Purpose

nag_lapack_zupgtr (f08gt) generates the complex unitary matrix QQ, which was determined by nag_lapack_zhptrd (f08gs) when reducing a Hermitian matrix to tridiagonal form.

Syntax

[q, info] = f08gt(uplo, n, ap, tau)
[q, info] = nag_lapack_zupgtr(uplo, n, ap, tau)

Description

nag_lapack_zupgtr (f08gt) is intended to be used after a call to nag_lapack_zhptrd (f08gs), which reduces a complex Hermitian matrix AA to real symmetric tridiagonal form TT by a unitary similarity transformation: A = QTQHA=QTQH. nag_lapack_zhptrd (f08gs) represents the unitary 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_zhptrd (f08gs).
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix QQ.
Constraint: n0n0.
3:     ap( : :) – complex 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_zhptrd (f08gs).
4:     tau( : :) – complex 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_zhptrd (f08gs).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

ldq work

Output Parameters

1:     q(ldq, : :) – complex 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 unitary 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 unitary 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 real floating point operations is approximately (16/3)n3163n3.
The real analogue of this function is nag_lapack_dopgtr (f08gf).

Example

function nag_lapack_zupgtr_example
uplo = 'L';
n = int64(4);
ap = [complex(-2.28);
      -4.33845594653213 + 0i;
      0.3278606760921924 - 0.1251226092264437i;
      -0.1412565637506947 - 0.366636483973957i;
      -0.1284569816493291 + 0i;
      -2.022594578622617 + 0i;
      -0.308321908008089 + 0.1763226364726777i;
      -0.1665932537524081 + 0i;
      -1.802322978338735 + 0i;
      -1.924949764598263 + 0i];
tau = [ 1.410284216766754 + 0.4679084045148932i;
      1.302420369434775 + 0.7853320742529579i;
      1.093973715923082 - 0.9955746786231597i];
[q, info] = nag_lapack_zupgtr(uplo, n, ap, tau)
 

q =

   1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -0.4103 - 0.4679i   0.0689 + 0.1780i   0.6583 + 0.3781i
   0.0000 + 0.0000i  -0.5209 + 0.0230i  -0.2576 - 0.7356i  -0.2313 + 0.2592i
   0.0000 + 0.0000i   0.0277 + 0.5832i   0.5956 - 0.0379i   0.0657 + 0.5465i


info =

                    0


function f08gt_example
uplo = 'L';
n = int64(4);
ap = [complex(-2.28);
      -4.33845594653213 + 0i;
      0.3278606760921924 - 0.1251226092264437i;
      -0.1412565637506947 - 0.366636483973957i;
      -0.1284569816493291 + 0i;
      -2.022594578622617 + 0i;
      -0.308321908008089 + 0.1763226364726777i;
      -0.1665932537524081 + 0i;
      -1.802322978338735 + 0i;
      -1.924949764598263 + 0i];
tau = [ 1.410284216766754 + 0.4679084045148932i;
      1.302420369434775 + 0.7853320742529579i;
      1.093973715923082 - 0.9955746786231597i];
[q, info] = f08gt(uplo, n, ap, tau)
 

q =

   1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -0.4103 - 0.4679i   0.0689 + 0.1780i   0.6583 + 0.3781i
   0.0000 + 0.0000i  -0.5209 + 0.0230i  -0.2576 - 0.7356i  -0.2313 + 0.2592i
   0.0000 + 0.0000i   0.0277 + 0.5832i   0.5956 - 0.0379i   0.0657 + 0.5465i


info =

                    0



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