SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER FACT, TRANS
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
$ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* SGTSVX uses the LU factorization to compute the solution to a real
* system of linear equations A * X = B or A**T * X = B,
* where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
* as A = L * U, where L is a product of permutation and unit lower
* bidiagonal matrices and U is upper triangular with nonzeros in
* only the main diagonal and first two superdiagonals.
*
* 2. If some U(i,i)=0, so that U is exactly singular, 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': DLF, DF, DUF, DU2, and IPIV contain the factored
* form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
* will not be modified.
* = 'N': The matrix will be copied to DLF, DF, and DUF
* and factored.
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* 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 matrix B. NRHS >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of A.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) superdiagonal elements of A.
*
* DLF (input or output) REAL array, dimension (N-1)
* If FACT = 'F', then DLF is an input argument and on entry
* contains the (n-1) multipliers that define the matrix L from
* the LU factorization of A as computed by SGTTRF.
*
* If FACT = 'N', then DLF is an output argument and on exit
* contains the (n-1) multipliers that define the matrix L from
* the LU factorization of 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 upper triangular
* matrix U from the LU factorization of A.
*
* If FACT = 'N', then DF is an output argument and on exit
* contains the n diagonal elements of the upper triangular
* matrix U from the LU factorization of A.
*
* DUF (input or output) REAL array, dimension (N-1)
* If FACT = 'F', then DUF is an input argument and on entry
* contains the (n-1) elements of the first superdiagonal of U.
*
* If FACT = 'N', then DUF is an output argument and on exit
* contains the (n-1) elements of the first superdiagonal of U.
*
* DU2 (input or output) REAL array, dimension (N-2)
* If FACT = 'F', then DU2 is an input argument and on entry
* contains the (n-2) elements of the second superdiagonal of
* U.
*
* If FACT = 'N', then DU2 is an output argument and on exit
* contains the (n-2) elements of the second superdiagonal of
* U.
*
* IPIV (input or output) INTEGER array, dimension (N)
* If FACT = 'F', then IPIV is an input argument and on entry
* contains the pivot indices from the LU factorization of A as
* computed by SGTTRF.
*
* If FACT = 'N', then IPIV is an output argument and on exit
* contains the pivot indices from the LU factorization of A;
* row i of the matrix was interchanged with row IPIV(i).
* IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
* a row interchange was not required.
*
* 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 or 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 estimate of 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 estimated 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). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* 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 (3*N)
*
* IWORK (workspace) INTEGER array, dimension (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: U(i,i) is exactly zero. The factorization
* has not been completed unless i = N, but the
* factor U is exactly singular, so the solution
* and error bounds could not be 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, NOTRAN
CHARACTER NORM
REAL ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANGT
EXTERNAL LSAME, SLAMCH, SLANGT
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGTCON, SGTRFS, SGTTRF, SGTTRS, SLACPY,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGTSVX', -INFO )
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the LU factorization of A.
*
CALL SCOPY( N, D, 1, DF, 1 )
IF( N.GT.1 ) THEN
CALL SCOPY( N-1, DL, 1, DLF, 1 )
CALL SCOPY( N-1, DU, 1, DUF, 1 )
END IF
CALL SGTTRF( N, DLF, DF, DUF, DU2, IPIV, 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.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = SLANGT( NORM, N, DL, D, DU )
*
* Compute the reciprocal of the condition number of A.
*
CALL SGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
$ IWORK, INFO )
*
* Compute the solution vectors X.
*
CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL SGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RETURN
*
* End of SGTSVX
*
END