sptsvx()

subroutine sptsvx	(	character	fact,
		integer	n,
		integer	nrhs,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( * )	df,
		real, dimension( * )	ef,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( ldx, * )	x,
		integer	ldx,
		real	rcond,
		real, dimension( * )	ferr,
		real, dimension( * )	berr,
		real, dimension( * )	work,
		integer	info )

SPTSVX computes the solution to system of linear equations A * X = B for PT matrices

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Purpose:

!>
!> SPTSVX uses the factorization A = L*D*L**T to compute the solution
!> to a real system of linear equations A*X = B, where A is an N-by-N
!> symmetric positive definite tridiagonal matrix and X and B are
!> N-by-NRHS matrices.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!> 

Description:

!>
!> The following steps are performed:
!>
!> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
!>    is a unit lower bidiagonal matrix and D is diagonal.  The
!>    factorization can also be regarded as having the form
!>    A = U**T*D*U.
!>
!> 2. If the leading principal minor of order i is not positive,
!>    then the routine returns with INFO = i. Otherwise, the factored
!>    form of A is used to estimate the condition number of the matrix
!>    A.  If the reciprocal of the condition number is less than machine
!>    precision, INFO = N+1 is returned as a warning, but the routine
!>    still goes on to solve for X and compute error bounds as
!>    described below.
!>
!> 3. The system of equations is solved for X using the factored form
!>    of A.
!>
!> 4. Iterative refinement is applied to improve the computed solution
!>    matrix and calculate error bounds and backward error estimates
!>    for it.
!> 

Parameters

[in]	FACT	!> FACT is CHARACTER*1 !> Specifies whether or not the factored form of A has been !> supplied on entry. !> = 'F': On entry, DF and EF contain the factored form of A. !> D, E, DF, and EF will not be modified. !> = 'N': The matrix A will be copied to DF and EF and !> factored. !>
[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the tridiagonal matrix A. !>
[in]	E	!> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix A. !>
[in,out]	DF	!> DF is REAL array, dimension (N) !> If FACT = 'F', then DF is an input argument and on entry !> contains the n diagonal elements of the diagonal matrix D !> from the L*D*L**T factorization of A. !> If FACT = 'N', then DF is an output argument and on exit !> contains the n diagonal elements of the diagonal matrix D !> from the L*D*L**T factorization of A. !>
[in,out]	EF	!> EF is REAL array, dimension (N-1) !> If FACT = 'F', then EF is an input argument and on entry !> contains the (n-1) subdiagonal elements of the unit !> bidiagonal factor L from the L*D*L**T factorization of A. !> If FACT = 'N', then EF is an output argument and on exit !> contains the (n-1) subdiagonal elements of the unit !> bidiagonal factor L from the L*D*L**T factorization of A. !>
[in]	B	!> B is REAL array, dimension (LDB,NRHS) !> The N-by-NRHS right hand side matrix B. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	X	!> X is REAL array, dimension (LDX,NRHS) !> If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	RCOND	!> RCOND is REAL !> The reciprocal condition number of the matrix A. If RCOND !> is less than the machine precision (in particular, if !> RCOND = 0), the matrix is singular to working precision. !> This condition is indicated by a return code of INFO > 0. !>
[out]	FERR	!> FERR is REAL array, dimension (NRHS) !> The forward error bound for each solution vector !> X(j) (the j-th column of the solution matrix X). !> If XTRUE is the true solution corresponding to X(j), FERR(j) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). !>
[out]	BERR	!> BERR is REAL array, dimension (NRHS) !> The componentwise relative backward error of each solution !> vector X(j) (i.e., the smallest relative change in any !> element of A or B that makes X(j) an exact solution). !>
[out]	WORK	!> WORK is REAL array, dimension (2*N) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, and i is !> <= N: the leading principal minor of order i of A !> is not positive, so the factorization could not !> be completed, and the solution has not been !> computed. RCOND = 0 is returned. !> = N+1: U is nonsingular, but RCOND is less than machine !> precision, meaning that the matrix is singular !> to working precision. Nevertheless, the !> solution and error bounds are computed because !> there are a number of situations where the !> computed solution can be more accurate than the !> value of RCOND would suggest. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 224 of file sptsvx.f.

*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          FACT
      INTEGER            INFO, LDB, LDX, N, NRHS
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   E( * ), EF( * ), FERR( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO
      parameter( zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOFACT
      REAL               ANORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANST
      EXTERNAL           lsame, slamch, slanst
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slacpy, sptcon, sptrfs, spttrf,
     $                   spttrs,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      nofact = lsame( fact, 'N' )
      IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( nrhs.LT.0 ) THEN
         info = -3
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -9
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -11
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTSVX', -info )
         RETURN
      END IF
*
      IF( nofact ) THEN
*
*        Compute the L*D*L**T (or U**T*D*U) factorization of A.
*
         CALL scopy( n, d, 1, df, 1 )
         IF( n.GT.1 )
     $      CALL scopy( n-1, e, 1, ef, 1 )
         CALL spttrf( n, df, ef, info )
*
*        Return if INFO is non-zero.
*
         IF( info.GT.0 )THEN
            rcond = zero
            RETURN
         END IF
      END IF
*
*     Compute the norm of the matrix A.
*
      anorm = slanst( '1', n, d, e )
*
*     Compute the reciprocal of the condition number of A.
*
      CALL sptcon( n, df, ef, anorm, rcond, work, info )
*
*     Compute the solution vectors X.
*
      CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
      CALL spttrs( n, nrhs, df, ef, x, ldx, info )
*
*     Use iterative refinement to improve the computed solutions and
*     compute error bounds and backward error estimates for them.
*
      CALL sptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr,
     $             work, info )
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( rcond.LT.slamch( 'Epsilon' ) )
     $   info = n + 1
*
      RETURN
*
*     End of SPTSVX
*

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