sspt21.f - Man Page

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

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

Univ. of Colorado Denver

NAG Ltd.

Definition at line 219 of file sspt21.f.