LAPACK 3.12.1
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

Download SGTSVX + dependencies [TGZ] [ZIP] [TXT]

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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 288 of file sgtsvx.f.

292*
293* -- LAPACK driver routine --
294* -- LAPACK is a software package provided by Univ. of Tennessee, --
295* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
296*
297* .. Scalar Arguments ..
298 CHARACTER FACT, TRANS
299 INTEGER INFO, LDB, LDX, N, NRHS
300 REAL RCOND
301* ..
302* .. Array Arguments ..
303 INTEGER IPIV( * ), IWORK( * )
304 REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
305 $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
306 $ FERR( * ), WORK( * ), X( LDX, * )
307* ..
308*
309* =====================================================================
310*
311* .. Parameters ..
312 REAL ZERO
313 parameter( zero = 0.0e+0 )
314* ..
315* .. Local Scalars ..
316 LOGICAL NOFACT, NOTRAN
317 CHARACTER NORM
318 REAL ANORM
319* ..
320* .. External Functions ..
321 LOGICAL LSAME
322 REAL SLAMCH, SLANGT
323 EXTERNAL lsame, slamch, slangt
324* ..
325* .. External Subroutines ..
326 EXTERNAL scopy, sgtcon, sgtrfs, sgttrf, sgttrs,
327 $ 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,
388 $ work,
389 $ iwork, info )
390*
391* Compute the solution vectors X.
392*
393 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
394 CALL sgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
395 $ info )
396*
397* Use iterative refinement to improve the computed solutions and
398* compute error bounds and backward error estimates for them.
399*
400 CALL sgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2,
401 $ ipiv,
402 $ b, ldb, x, ldx, ferr, berr, work, iwork, info )
403*
404* Set INFO = N+1 if the matrix is singular to working precision.
405*
406 IF( rcond.LT.slamch( 'Epsilon' ) )
407 $ info = n + 1
408*
409 RETURN
410*
411* End of SGTSVX
412*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
scopy
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
sgtcon
subroutine sgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info)
SGTCON
Definition sgtcon.f:144
sgtrfs
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:208
sgttrf
subroutine sgttrf(n, dl, d, du, du2, ipiv, info)
SGTTRF
Definition sgttrf.f:122
sgttrs
subroutine sgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
SGTTRS
Definition sgttrs.f:137
slacpy
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:101
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
slangt
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:104
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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