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

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

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

Purpose:
``` CGTSVX uses the LU factorization to compute the solution to a complex
system of linear equations A * X = B, A**T * X = B, or A**H * 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)``` [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 COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is COMPLEX array, dimension (N) The n diagonal elements of A.``` [in] DU ``` DU is COMPLEX array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in,out] DLF ``` DLF is COMPLEX 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 CGTTRF. 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CGTTRF. 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (2*N)` [out] RWORK ` RWORK is REAL 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 296 of file cgtsvx.f.

296 *
297 * -- LAPACK driver routine (version 3.4.2) --
298 * -- LAPACK is a software package provided by Univ. of Tennessee, --
299 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
300 * September 2012
301 *
302 * .. Scalar Arguments ..
303  CHARACTER fact, trans
304  INTEGER info, ldb, ldx, n, nrhs
305  REAL rcond
306 * ..
307 * .. Array Arguments ..
308  INTEGER ipiv( * )
309  REAL berr( * ), ferr( * ), rwork( * )
310  COMPLEX b( ldb, * ), d( * ), df( * ), dl( * ),
311  \$ dlf( * ), du( * ), du2( * ), duf( * ),
312  \$ work( * ), x( ldx, * )
313 * ..
314 *
315 * =====================================================================
316 *
317 * .. Parameters ..
318  REAL zero
319  parameter ( zero = 0.0e+0 )
320 * ..
321 * .. Local Scalars ..
322  LOGICAL nofact, notran
323  CHARACTER norm
324  REAL anorm
325 * ..
326 * .. External Functions ..
327  LOGICAL lsame
328  REAL clangt, slamch
329  EXTERNAL lsame, clangt, slamch
330 * ..
331 * .. External Subroutines ..
332  EXTERNAL ccopy, cgtcon, cgtrfs, cgttrf, cgttrs, clacpy,
333  \$ xerbla
334 * ..
335 * .. Intrinsic Functions ..
336  INTRINSIC max
337 * ..
338 * .. Executable Statements ..
339 *
340  info = 0
341  nofact = lsame( fact, 'N' )
342  notran = lsame( trans, 'N' )
343  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
344  info = -1
345  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
346  \$ lsame( trans, 'C' ) ) THEN
347  info = -2
348  ELSE IF( n.LT.0 ) THEN
349  info = -3
350  ELSE IF( nrhs.LT.0 ) THEN
351  info = -4
352  ELSE IF( ldb.LT.max( 1, n ) ) THEN
353  info = -14
354  ELSE IF( ldx.LT.max( 1, n ) ) THEN
355  info = -16
356  END IF
357  IF( info.NE.0 ) THEN
358  CALL xerbla( 'CGTSVX', -info )
359  RETURN
360  END IF
361 *
362  IF( nofact ) THEN
363 *
364 * Compute the LU factorization of A.
365 *
366  CALL ccopy( n, d, 1, df, 1 )
367  IF( n.GT.1 ) THEN
368  CALL ccopy( n-1, dl, 1, dlf, 1 )
369  CALL ccopy( n-1, du, 1, duf, 1 )
370  END IF
371  CALL cgttrf( n, dlf, df, duf, du2, ipiv, info )
372 *
373 * Return if INFO is non-zero.
374 *
375  IF( info.GT.0 )THEN
376  rcond = zero
377  RETURN
378  END IF
379  END IF
380 *
381 * Compute the norm of the matrix A.
382 *
383  IF( notran ) THEN
384  norm = '1'
385  ELSE
386  norm = 'I'
387  END IF
388  anorm = clangt( norm, n, dl, d, du )
389 *
390 * Compute the reciprocal of the condition number of A.
391 *
392  CALL cgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
393  \$ info )
394 *
395 * Compute the solution vectors X.
396 *
397  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
398  CALL cgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
399  \$ info )
400 *
401 * Use iterative refinement to improve the computed solutions and
402 * compute error bounds and backward error estimates for them.
403 *
404  CALL cgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
405  \$ b, ldb, x, ldx, ferr, berr, work, rwork, info )
406 *
407 * Set INFO = N+1 if the matrix is singular to working precision.
408 *
409  IF( rcond.LT.slamch( 'Epsilon' ) )
410  \$ info = n + 1
411 *
412  RETURN
413 *
414 * End of CGTSVX
415 *
subroutine cgttrf(N, DL, D, DU, DU2, IPIV, INFO)
CGTTRF
Definition: cgttrf.f:126
subroutine cgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)
CGTCON
Definition: cgtcon.f:143
subroutine cgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
CGTTRS
Definition: cgttrs.f:140
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function clangt(NORM, N, DL, D, DU)
CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clangt.f:108
subroutine cgtrfs(TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CGTRFS
Definition: cgtrfs.f:212
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:52
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

Here is the call graph for this function:

Here is the caller graph for this function: