TESTING/EIG/dspt21.f(3)

NAME
TESTING/EIG/dspt21.f

SYNOPSIS

  Functions/Subroutines
subroutine dspt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, result)
DSPT21

Function/Subroutine Documentation

  subroutine dspt21 (integer itype, character uplo, integer n, integer kband, double precision, dimension( * ) ap, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( * ) vp, double precision, dimension( * ) tau, double precision, dimension( * ) work, double precision, dimension( 2 ) result)

DSPT21

Purpose:

     DSPT21  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, DSPT21 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 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

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

    E

              E is 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

              U is 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

              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 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N**2+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

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 219 of file dspt21.f.

Author

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