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
    !>  and whose scaling constants are in
     we shall use the
    !> letter  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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