LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
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sspt21.f File Reference

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Functions/Subroutines

subroutine sspt21 (ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, TAU, WORK, RESULT)
 SSPT21 More...
 

Function/Subroutine Documentation

subroutine sspt21 ( integer  ITYPE,
character  UPLO,
integer  N,
integer  KBAND,
real, dimension( * )  AP,
real, dimension( * )  D,
real, dimension( * )  E,
real, dimension( ldu, * )  U,
integer  LDU,
real, dimension( * )  VP,
real, dimension( * )  TAU,
real, dimension( * )  WORK,
real, dimension( 2 )  RESULT 
)

SSPT21

Purpose:
 SSPT21  generally checks a decomposition of the form

         A = U S U'

 where ' 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' | / ( |A| n ulp ) *andC>         RESULT(2) = | I - UU' | / ( n ulp )

 If ITYPE=2, then:

         RESULT(1) = | A - V S V' | / ( |A| n ulp )

 If ITYPE=3, then:

         RESULT(1) = | I - VU' | / ( 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)'

    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)'

    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
[in]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' | / ( |A| n ulp )   *andC>             RESULT(2) = | I - UU' | / ( n ulp )

          2: U expressed as a product V of Housholder transformations:
             RESULT(1) = | A - V S V' | / ( |A| n ulp )

          3: U expressed both as a dense orthogonal matrix and
             as a product of Housholder transformations:
             RESULT(1) = | I - VU' | / ( n ulp )
[in]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.
[in]N
          N is INTEGER
          The size of the matrix.  If it is zero, SSPT21 does nothing.
          It must be at least zero.
[in]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.
[in]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.
[in]D
          D is REAL array, dimension (N)
          The diagonal of the (symmetric tri-) diagonal matrix.
[in]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.
[in]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.
[in]LDU
          LDU is INTEGER
          The leading dimension of U.  LDU must be at least N and
          at least 1.
[in]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.
[in]TAU
          TAU is REAL array, dimension (N)
          If ITYPE >= 2, then TAU(j) is the scalar factor of
          v(j) v(j)' in the Householder transformation H(j) of
          the product  U = H(1)...H(n-2)
          If ITYPE < 2, then TAU is not referenced.
[out]WORK
          WORK is REAL array, dimension (N**2+N)
          Workspace.
[out]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.
Date
November 2011

Definition at line 219 of file sspt21.f.

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