DSYT21
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DSYT21 generally checks a decomposition of the form
A = U S U'
where ' means transpose, A is symmetric, 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 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 )
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)' and each
vector v(j) has its first j elements 0 and the remaining n-j elements
stored in V(j+1:n,j).
A = U S U'
where ' means transpose, A is symmetric, 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 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 )
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)' and each
vector v(j) has its first j elements 0 and the remaining n-j elements
stored in V(j+1:n,j).
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', 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 |
(input) INTEGER
The size of the matrix. If it is zero, DSYT21 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. |
| A |
(input) DOUBLE PRECISION array, dimension (LDA, N)
The original (unfactored) matrix. It is assumed to be
symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. |
| LDA |
(input) INTEGER
The leading dimension of A. It must be at least 1
and at least N. |
| 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. |
| V |
(input) DOUBLE PRECISION array, dimension (LDV, N)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the orthogonal 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 |
(input) INTEGER
The leading dimension of V. LDV must be at least N and
at least 1. |
| 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 (2*N**2)
|
| 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. |