TESTING/EIG/zhpt21.f(3)

NAME
TESTING/EIG/zhpt21.f

SYNOPSIS

  Functions/Subroutines
subroutine zhpt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, rwork, result)
ZHPT21

Function/Subroutine Documentation

  subroutine zhpt21 (integer itype, character uplo, integer n, integer kband, complex*16, dimension( * ) ap, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( * ) vp, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, double precision, dimension( 2 ) result)

ZHPT21

Purpose:

     ZHPT21  generally checks a decomposition of the form
             A = U S U**H
     where **H means conjugate transpose, A is hermitian, U is
     unitary, and S is diagonal (if KBAND=0) or (real) 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**H | / ( |A| n ulp ) and
             RESULT(2) = | I - U U**H | / ( n ulp )
     If ITYPE=2, then:
             RESULT(1) = | A - V S V**H | / ( |A| n ulp )
     If ITYPE=3, then:
             RESULT(1) = | I - U V**H | / ( 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)**H
        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)**H
        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 unitary matrix:
                 RESULT(1) = | A - U S U**H | / ( |A| n ulp )   and
                 RESULT(2) = | I - U U**H | / ( n ulp )
              2: U expressed as a product V of Housholder transformations:
                 RESULT(1) = | A - V S V**H | / ( |A| n ulp )
              3: U expressed both as a dense unitary matrix and
                 as a product of Housholder transformations:
                 RESULT(1) = | I - U V**H | / ( n ulp )

    UPLO

              UPLO is 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

              N is INTEGER
              The size of the matrix.  If it is zero, ZHPT21 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 COMPLEX*16 array, dimension (N*(N+1)/2)
              The original (unfactored) matrix.  It is assumed to be
              hermitian, 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)
              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 COMPLEX*16 array, dimension (LDU, N)
              If ITYPE=1 or 3, this contains the unitary 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 unitary 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 COMPLEX*16 array, dimension (N)
              If ITYPE >= 2, then TAU(j) is the scalar factor of
              v(j) v(j)**H 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 COMPLEX*16 array, dimension (N**2)
              Workspace.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (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 226 of file zhpt21.f.

Author

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