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NAG Toolbox: nag_lapack_zhetrd (f08fs)

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 AA to real symmetric tridiagonal form TT by a unitary similarity transformation: A = QTQHA=QTQH.
The matrix QQ is not formed explicitly but is represented as a product of n1n-1 elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with QQ in this representation (see Section [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 AA is stored.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     a(lda, : :) – complex 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 Hermitian matrix AA.
  • If uplo = 'U'uplo='U', the upper triangular part of aa must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', the lower triangular part of aa 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 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, : :) – complex 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 tridiagonal matrix TT and details of the unitary matrix QQ as specified by uplo.
2:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
The diagonal elements of the tridiagonal matrix TT.
3:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
The off-diagonal elements of the tridiagonal matrix TT.
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 unitary matrix QQ.
5:     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: 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 TT 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 TT 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 and eigenvectors.

Further Comments

The total number of real floating point operations is approximately (16/3) n3 163 n3 .
To form the unitary matrix QQ nag_lapack_zhetrd (f08fs) may be followed by a call to nag_lapack_zungtr (f08ft):
[a, info] = f08ft(uplo, a, tau);
To apply QQ to an nn by pp complex matrix CC 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 QCQC.
The real analogue of this function is nag_lapack_dsytrd (f08fe).

Example

function nag_lapack_zhetrd_example
uplo = 'L';
a = [complex(-2.28),  0 + 0i,  0 + 0i,  0 + 0i;
      1.78 + 2.03i,  -1.12 + 0i,  0 + 0i,  0 + 0i;
      2.26 - 0.1i,  0.01 - 0.43i,  -0.37 + 0i,  0 + 0i;
      -0.12 - 2.53i,  -1.07 - 0.86i,  2.31 + 0.92i,  -0.73 + 0i];
[aOut, d, e, tau, info] = nag_lapack_zhetrd(uplo, a)
 

aOut =

  -2.2800 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
  -4.3385 + 0.0000i  -0.1285 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.3279 - 0.1251i  -2.0226 + 0.0000i  -0.1666 + 0.0000i   0.0000 + 0.0000i
  -0.1413 - 0.3666i  -0.3083 + 0.1763i  -1.8023 + 0.0000i  -1.9249 + 0.0000i


d =

   -2.2800
   -0.1285
   -0.1666
   -1.9249


e =

   -4.3385
   -2.0226
   -1.8023


tau =

   1.4103 + 0.4679i
   1.3024 + 0.7853i
   1.0940 - 0.9956i


info =

                    0


function f08fs_example
uplo = 'L';
a = [complex(-2.28),  0 + 0i,  0 + 0i,  0 + 0i;
      1.78 + 2.03i,  -1.12 + 0i,  0 + 0i,  0 + 0i;
      2.26 - 0.1i,  0.01 - 0.43i,  -0.37 + 0i,  0 + 0i;
      -0.12 - 2.53i,  -1.07 - 0.86i,  2.31 + 0.92i,  -0.73 + 0i];
[aOut, d, e, tau, info] = f08fs(uplo, a)
 

aOut =

  -2.2800 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
  -4.3385 + 0.0000i  -0.1285 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.3279 - 0.1251i  -2.0226 + 0.0000i  -0.1666 + 0.0000i   0.0000 + 0.0000i
  -0.1413 - 0.3666i  -0.3083 + 0.1763i  -1.8023 + 0.0000i  -1.9249 + 0.0000i


d =

   -2.2800
   -0.1285
   -0.1666
   -1.9249


e =

   -4.3385
   -2.0226
   -1.8023


tau =

   1.4103 + 0.4679i
   1.3024 + 0.7853i
   1.0940 - 0.9956i


info =

                    0



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