# sspt21.f - Man Page

TESTING/EIG/sspt21.f

## Synopsis

### Functions/Subroutines

subroutine sspt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, result)
SSPT21

## Function/Subroutine Documentation

### subroutine sspt21 (integer itype, character uplo, integer n, integer kband, real, dimension( * ) ap, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldu, * ) u, integer ldu, real, dimension( * ) vp, real, dimension( * ) tau, real, dimension( * ) work, real, dimension( 2 ) result)

SSPT21

Purpose:

``` SSPT21  generally checks a decomposition of the form

A = U S U**T

where **T means transpose, A is symmetric (stored in packed format), U
is orthogonal, and S is diagonal (if KBAND=0) or 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**T | / ( |A| n ulp ) and
RESULT(2) = | I - U U**T | / ( n ulp )

If ITYPE=2, then:

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

If ITYPE=3, then:

RESULT(1) = | I - V U**T | / ( 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)**T

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)**T

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 orthogonal matrix:
RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
RESULT(2) = | I - U U**T | / ( n ulp )

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

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

UPLO

```          UPLO is CHARACTER
If UPLO='U', AP and VP are considered to contain the upper
triangle of A and V.
If UPLO='L', AP and VP are considered to contain the lower
triangle of A and V.```

N

```          N is INTEGER
The size of the matrix.  If it is zero, SSPT21 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 REAL array, dimension (N*(N+1)/2)
The original (unfactored) matrix.  It is assumed to be
symmetric, 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 REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.```

E

```          E is REAL 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 REAL array, dimension (LDU, N)
If ITYPE=1 or 3, this contains the orthogonal 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 REAL array, dimension (N*(N+1)/2)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the orthogonal 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 REAL array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)**T 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 REAL array, dimension (N**2+N)
Workspace.```

RESULT

```          RESULT is REAL 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 219 of file sspt21.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sspt21(3) is an alias of sspt21.f(3).

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