# zhet21.f - Man Page

TESTING/EIG/zhet21.f

## Synopsis

### Functions/Subroutines

subroutine zhet21 (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv, tau, work, rwork, result)
ZHET21

## Function/Subroutine Documentation

### subroutine zhet21 (integer itype, character uplo, integer n, integer kband, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, double precision, dimension( 2 ) result)

ZHET21

Purpose:

ZHET21 generally checks a decomposition of the form

A = U S U**H

where **H means conjugate transpose, A is hermitian, U is unitary, and
S is diagonal (if KBAND=0) or (real) symmetric tridiagonal (if
KBAND=1).

If ITYPE=1, then U is represented as a dense matrix; otherwise U is
expressed as a product of Householder transformations, whose vectors
are stored in the array 'V' and whose scaling constants are in 'TAU'.
We shall use the letter 'V' to refer to the product of Householder
transformations (which should be equal to U).

Specifically, if ITYPE=1, then:

RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
RESULT(2) = | I - U U**H | / ( n ulp )

If ITYPE=2, then:

RESULT(1) = | A - V S V**H | / ( |A| n ulp )

If ITYPE=3, then:

RESULT(1) = | I - U V**H | / ( n ulp )

For ITYPE > 1, the transformation U is expressed as a product
V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)**H and each
vector v(j) has its first j elements 0 and the remaining n-j elements
stored in V(j+1:n,j).
Parameters

ITYPE

ITYPE is INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense unitary matrix:
RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
RESULT(2) = | I - U U**H | / ( n ulp )

2: U expressed as a product V of Housholder transformations:
RESULT(1) = | A - V S V**H | / ( |A| n ulp )

3: U expressed both as a dense unitary matrix and
as a product of Housholder transformations:
RESULT(1) = | I - U V**H | / ( n ulp )

UPLO

UPLO is CHARACTER
If UPLO='U', the upper triangle of A and V will be used and
the (strictly) lower triangle will not be referenced.
If UPLO='L', the lower triangle of A and V will be used and
the (strictly) upper triangle will not be referenced.

N

N is INTEGER
The size of the matrix.  If it is zero, ZHET21 does nothing.
It must be at least zero.

KBAND

KBAND is INTEGER
The bandwidth of the matrix.  It may only be zero or one.
If zero, then S is diagonal, and E is not referenced.  If
one, then S is symmetric tri-diagonal.

A

A is COMPLEX*16 array, dimension (LDA, N)
The original (unfactored) matrix.  It is assumed to be
hermitian, and only the upper (UPLO='U') or only the lower
(UPLO='L') will be referenced.

LDA

LDA is INTEGER
The leading dimension of A.  It must be at least 1
and at least N.

D

D is DOUBLE PRECISION array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.

E

E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal of the (symmetric tri-) diagonal matrix.
E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
(3,2) element, etc.
Not referenced if KBAND=0.

U

U is COMPLEX*16 array, dimension (LDU, N)
If ITYPE=1 or 3, this contains the unitary matrix in
the decomposition, expressed as a dense matrix.  If ITYPE=2,
then it is not referenced.

LDU

LDU is INTEGER
The leading dimension of U.  LDU must be at least N and
at least 1.

V

V is COMPLEX*16 array, dimension (LDV, N)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the unitary matrix
in the decomposition.  If UPLO='L', then the vectors are in
the lower triangle, if UPLO='U', then in the upper
triangle.
*NOTE* If ITYPE=2 or 3, V is modified and restored.  The
subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
is set to one, and later reset to its original value, during
the course of the calculation.
If ITYPE=1, then it is neither referenced nor modified.

LDV

LDV is INTEGER
The leading dimension of V.  LDV must be at least N and
at least 1.

TAU

TAU is COMPLEX*16 array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)**H in the Householder transformation H(j) of
the product  U = H(1)...H(n-2)
If ITYPE < 2, then TAU is not referenced.

WORK

WORK is COMPLEX*16 array, dimension (2*N**2)

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

RESULT

RESULT is DOUBLE PRECISION array, dimension (2)
The values computed by the two tests described above.  The
values are currently limited to 1/ulp, to avoid overflow.
RESULT(1) is always modified.  RESULT(2) is modified only
if ITYPE=1.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 212 of file zhet21.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zhet21(3) is an alias of zhet21.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK