LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sgtsvx()

 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

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.```

Definition at line 290 of file sgtsvx.f.

293*
294* -- LAPACK driver routine --
295* -- LAPACK is a software package provided by Univ. of Tennessee, --
296* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
297*
298* .. Scalar Arguments ..
299 CHARACTER FACT, TRANS
300 INTEGER INFO, LDB, LDX, N, NRHS
301 REAL RCOND
302* ..
303* .. Array Arguments ..
304 INTEGER IPIV( * ), IWORK( * )
305 REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
306 \$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
307 \$ FERR( * ), WORK( * ), X( LDX, * )
308* ..
309*
310* =====================================================================
311*
312* .. Parameters ..
313 REAL ZERO
314 parameter( zero = 0.0e+0 )
315* ..
316* .. Local Scalars ..
317 LOGICAL NOFACT, NOTRAN
318 CHARACTER NORM
319 REAL ANORM
320* ..
321* .. External Functions ..
322 LOGICAL LSAME
323 REAL SLAMCH, SLANGT
324 EXTERNAL lsame, slamch, slangt
325* ..
326* .. External Subroutines ..
327 EXTERNAL scopy, sgtcon, sgtrfs, sgttrf, sgttrs, slacpy,
328 \$ xerbla
329* ..
330* .. Intrinsic Functions ..
331 INTRINSIC max
332* ..
333* .. Executable Statements ..
334*
335 info = 0
336 nofact = lsame( fact, 'N' )
337 notran = lsame( trans, 'N' )
338 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
339 info = -1
340 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
341 \$ lsame( trans, 'C' ) ) THEN
342 info = -2
343 ELSE IF( n.LT.0 ) THEN
344 info = -3
345 ELSE IF( nrhs.LT.0 ) THEN
346 info = -4
347 ELSE IF( ldb.LT.max( 1, n ) ) THEN
348 info = -14
349 ELSE IF( ldx.LT.max( 1, n ) ) THEN
350 info = -16
351 END IF
352 IF( info.NE.0 ) THEN
353 CALL xerbla( 'SGTSVX', -info )
354 RETURN
355 END IF
356*
357 IF( nofact ) THEN
358*
359* Compute the LU factorization of A.
360*
361 CALL scopy( n, d, 1, df, 1 )
362 IF( n.GT.1 ) THEN
363 CALL scopy( n-1, dl, 1, dlf, 1 )
364 CALL scopy( n-1, du, 1, duf, 1 )
365 END IF
366 CALL sgttrf( n, dlf, df, duf, du2, ipiv, info )
367*
368* Return if INFO is non-zero.
369*
370 IF( info.GT.0 )THEN
371 rcond = zero
372 RETURN
373 END IF
374 END IF
375*
376* Compute the norm of the matrix A.
377*
378 IF( notran ) THEN
379 norm = '1'
380 ELSE
381 norm = 'I'
382 END IF
383 anorm = slangt( norm, n, dl, d, du )
384*
385* Compute the reciprocal of the condition number of A.
386*
387 CALL sgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
388 \$ iwork, info )
389*
390* Compute the solution vectors X.
391*
392 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
393 CALL sgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
394 \$ info )
395*
396* Use iterative refinement to improve the computed solutions and
397* compute error bounds and backward error estimates for them.
398*
399 CALL sgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
400 \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
401*
402* Set INFO = N+1 if the matrix is singular to working precision.
403*
404 IF( rcond.LT.slamch( 'Epsilon' ) )
405 \$ info = n + 1
406*
407 RETURN
408*
409* End of SGTSVX
410*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info)
SGTCON
Definition sgtcon.f:146
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:209
subroutine sgttrf(n, dl, d, du, du2, ipiv, info)
SGTTRF
Definition sgttrf.f:124
subroutine sgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
SGTTRS
Definition sgttrs.f:138
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:103
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
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:106
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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