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SUBROUTINE SPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. 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, * )
* ..
*
* 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 i-by-i principal minor is not positive definite,
* 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.
*
* Arguments
* =========
*
* FACT (input) 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.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the tridiagonal matrix A.
*
* DF (input or output) 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.
*
* EF (input or output) 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.
*
* B (input) REAL array, dimension (LDB,NRHS)
* The N-by-NRHS right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) REAL array, dimension (LDX,NRHS)
* If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) 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.
*
* FERR (output) 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).
*
* BERR (output) 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).
*
* WORK (workspace) REAL array, dimension (2*N)
*
* INFO (output) 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 minor of order i of A is
* not positive definite, 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.
*
* =====================================================================
*
* .. 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' (or U'*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
*
END
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