.TH "TESTING/EIG/zhpt21.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
TESTING/EIG/zhpt21.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBzhpt21\fP (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, rwork, result)"
.br
.RI "\fBZHPT21\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBZHPT21\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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 ) \&.)
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIITYPE\fP 
.PP
.nf
          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 )
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The size of the matrix\&.  If it is zero, ZHPT21 does nothing\&.
          It must be at least zero\&.
.fi
.PP
.br
\fIKBAND\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIAP\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
          The diagonal of the (symmetric tri-) diagonal matrix\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of U\&.  LDU must be at least N and
          at least 1\&.
.fi
.PP
.br
\fIVP\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (N**2)
          Workspace\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
          Workspace\&.
.fi
.PP
.br
\fIRESULT\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB226\fP of file \fBzhpt21\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.