# zhpt21.f - Man Page

TESTING/EIG/zhpt21.f

## Synopsis

### Functions/Subroutines

subroutine zhpt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, rwork, result)
ZHPT21

## Function/Subroutine Documentation

### subroutine zhpt21 (integer itype, character uplo, integer n, integer kband, complex*16, dimension( * ) ap, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( * ) vp, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, double precision, dimension( 2 ) result)

ZHPT21

Purpose:

``` ZHPT21  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 the 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 )

Packed storage means that, for example, if UPLO='U', then the columns
of the upper triangle of A are stored one after another, so that
A(1,j+1) immediately follows A(j,j) in the array AP.  Similarly, if
UPLO='L', then the columns of the lower triangle of A are stored one
after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
in the array AP.  This means that A(i,j) is stored in:

AP( i + j*(j-1)/2 )                 if UPLO='U'

AP( i + (2*n-j)*(j-1)/2 )           if UPLO='L'

The array VP bears the same relation to the matrix V that A does to
AP.

For ITYPE > 1, the transformation U is expressed as a product
of Householder transformations:

If UPLO='U', then  V = H(n-1)...H(1),  where

H(j) = I  -  tau(j) v(j) v(j)**H

and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
(i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
the j-th element is 1, and the last n-j elements are 0.

If UPLO='L', then  V = H(1)...H(n-1),  where

H(j) = I  -  tau(j) v(j) v(j)**H

and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
(j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)```
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, ZHPT21 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.```

AP

```          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The original (unfactored) matrix.  It is assumed to be
hermitian, and contains the columns of just the upper
triangle (UPLO='U') or only the lower triangle (UPLO='L'),
packed one after another.```

D

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

E

```          E is DOUBLE PRECISION array, dimension (N)
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.```

VP

```          VP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the unitary matrix
in the decomposition, as described in purpose.
*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.```

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 (N**2)
Workspace.```

RWORK

```          RWORK is DOUBLE PRECISION array, dimension (N)
Workspace.```

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

NAG Ltd.

Definition at line 226 of file zhpt21.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zhpt21(3) is an alias of zhpt21.f(3).

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