!>
!> 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 ) .)
!> 

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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.