LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine sgtsvx ( character FACT, character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DLF, real, dimension( * ) DF, real, dimension( * ) DUF, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

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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.```
Parameters
 [in] FACT ``` FACT is 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.``` [in] TRANS ``` TRANS is 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)``` [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 matrix B. NRHS >= 0.``` [in] DL ``` DL is REAL array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of A.``` [in] DU ``` DU is REAL array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in,out] DLF ``` DLF is 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.``` [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 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.``` [in,out] DUF ``` DUF is 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.``` [in,out] DU2 ``` DU2 is 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.``` [in,out] IPIV ``` IPIV is 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.``` [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 or 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 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.``` [out] FERR ``` FERR is 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.``` [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 (3*N)` [out] IWORK ` IWORK is INTEGER array, dimension (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: 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.```
Date
September 2012

Definition at line 295 of file sgtsvx.f.

295 *
296 * -- LAPACK driver routine (version 3.4.2) --
297 * -- LAPACK is a software package provided by Univ. of Tennessee, --
298 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
299 * September 2012
300 *
301 * .. Scalar Arguments ..
302  CHARACTER fact, trans
303  INTEGER info, ldb, ldx, n, nrhs
304  REAL rcond
305 * ..
306 * .. Array Arguments ..
307  INTEGER ipiv( * ), iwork( * )
308  REAL b( ldb, * ), berr( * ), d( * ), df( * ),
309  \$ dl( * ), dlf( * ), du( * ), du2( * ), duf( * ),
310  \$ ferr( * ), work( * ), x( ldx, * )
311 * ..
312 *
313 * =====================================================================
314 *
315 * .. Parameters ..
316  REAL zero
317  parameter ( zero = 0.0e+0 )
318 * ..
319 * .. Local Scalars ..
320  LOGICAL nofact, notran
321  CHARACTER norm
322  REAL anorm
323 * ..
324 * .. External Functions ..
325  LOGICAL lsame
326  REAL slamch, slangt
327  EXTERNAL lsame, slamch, slangt
328 * ..
329 * .. External Subroutines ..
330  EXTERNAL scopy, sgtcon, sgtrfs, sgttrf, sgttrs, slacpy,
331  \$ xerbla
332 * ..
333 * .. Intrinsic Functions ..
334  INTRINSIC max
335 * ..
336 * .. Executable Statements ..
337 *
338  info = 0
339  nofact = lsame( fact, 'N' )
340  notran = lsame( trans, 'N' )
341  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
342  info = -1
343  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
344  \$ lsame( trans, 'C' ) ) THEN
345  info = -2
346  ELSE IF( n.LT.0 ) THEN
347  info = -3
348  ELSE IF( nrhs.LT.0 ) THEN
349  info = -4
350  ELSE IF( ldb.LT.max( 1, n ) ) THEN
351  info = -14
352  ELSE IF( ldx.LT.max( 1, n ) ) THEN
353  info = -16
354  END IF
355  IF( info.NE.0 ) THEN
356  CALL xerbla( 'SGTSVX', -info )
357  RETURN
358  END IF
359 *
360  IF( nofact ) THEN
361 *
362 * Compute the LU factorization of A.
363 *
364  CALL scopy( n, d, 1, df, 1 )
365  IF( n.GT.1 ) THEN
366  CALL scopy( n-1, dl, 1, dlf, 1 )
367  CALL scopy( n-1, du, 1, duf, 1 )
368  END IF
369  CALL sgttrf( n, dlf, df, duf, du2, ipiv, info )
370 *
371 * Return if INFO is non-zero.
372 *
373  IF( info.GT.0 )THEN
374  rcond = zero
375  RETURN
376  END IF
377  END IF
378 *
379 * Compute the norm of the matrix A.
380 *
381  IF( notran ) THEN
382  norm = '1'
383  ELSE
384  norm = 'I'
385  END IF
386  anorm = slangt( norm, n, dl, d, du )
387 *
388 * Compute the reciprocal of the condition number of A.
389 *
390  CALL sgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
391  \$ iwork, info )
392 *
393 * Compute the solution vectors X.
394 *
395  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
396  CALL sgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
397  \$ info )
398 *
399 * Use iterative refinement to improve the computed solutions and
400 * compute error bounds and backward error estimates for them.
401 *
402  CALL sgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
403  \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
404 *
405 * Set INFO = N+1 if the matrix is singular to working precision.
406 *
407  IF( rcond.LT.slamch( 'Epsilon' ) )
408  \$ info = n + 1
409 *
410  RETURN
411 *
412 * End of SGTSVX
413 *
subroutine sgtrfs(TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGTRFS
Definition: sgtrfs.f:211
subroutine sgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGTCON
Definition: sgtcon.f:148
real function slangt(NORM, N, DL, D, DU)
SLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slangt.f:108
subroutine sgttrf(N, DL, D, DU, DU2, IPIV, INFO)
SGTTRF
Definition: sgttrf.f:126
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
SGTTRS
Definition: sgttrs.f:140
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53

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