DSPT21
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DSPT21 generally checks a decomposition of the form
A = U S U'
where ' 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' | / ( |A| n ulp ) *and*
RESULT(2) = | I - UU' | / ( n ulp )
If ITYPE=2, then:
RESULT(1) = | A - V S V' | / ( |A| n ulp )
If ITYPE=3, then:
RESULT(1) = | I - VU' | / ( 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)'
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)'
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 ) .)
A = U S U'
where ' 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' | / ( |A| n ulp ) *and*
RESULT(2) = | I - UU' | / ( n ulp )
If ITYPE=2, then:
RESULT(1) = | A - V S V' | / ( |A| n ulp )
If ITYPE=3, then:
RESULT(1) = | I - VU' | / ( 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)'
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)'
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 ) .)
Arguments
| ITYPE |
(input) INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense orthogonal matrix: RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* RESULT(2) = | I - UU' | / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) = | A - V S V' | / ( |A| n ulp ) 3: U expressed both as a dense orthogonal matrix and as a product of Housholder transformations: RESULT(1) = | I - VU' | / ( n ulp ) |
| UPLO |
(input) 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 |
(input) INTEGER
The size of the matrix. If it is zero, DSPT21 does nothing.
It must be at least zero. |
| KBAND |
(input) 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 |
(input) DOUBLE PRECISION 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 |
(input) DOUBLE PRECISION array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
|
| E |
(input) 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 |
(input) DOUBLE PRECISION 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 |
(input) INTEGER
The leading dimension of U. LDU must be at least N and
at least 1. |
| VP |
(input) 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 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 |
(input) DOUBLE PRECISION array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)' in the Householder transformation H(j) of the product U = H(1)...H(n-2) If ITYPE < 2, then TAU is not referenced. |
| WORK |
(workspace) DOUBLE PRECISION array, dimension (N**2+N)
Workspace.
|
| RESULT |
(output) 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. |