LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zgtsvx()

 subroutine zgtsvx ( character FACT, character TRANS, integer N, integer NRHS, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DLF, complex*16, dimension( * ) DF, complex*16, dimension( * ) DUF, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO )

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

Purpose:
``` ZGTSVX 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*16 array, dimension (N-1) The (n-1) subdiagonal elements of A.``` [in] D ``` D is COMPLEX*16 array, dimension (N) The n diagonal elements of A.``` [in] DU ``` DU is COMPLEX*16 array, dimension (N-1) The (n-1) superdiagonal elements of A.``` [in,out] DLF ``` DLF is COMPLEX*16 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 ZGTTRF. 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*16 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*16 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*16 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 ZGTTRF. 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (2*N)` [out] RWORK ` RWORK is DOUBLE PRECISION 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 291 of file zgtsvx.f.

294 *
295 * -- LAPACK driver routine --
296 * -- LAPACK is a software package provided by Univ. of Tennessee, --
297 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
298 *
299 * .. Scalar Arguments ..
300  CHARACTER FACT, TRANS
301  INTEGER INFO, LDB, LDX, N, NRHS
302  DOUBLE PRECISION RCOND
303 * ..
304 * .. Array Arguments ..
305  INTEGER IPIV( * )
306  DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
307  COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
308  \$ DLF( * ), DU( * ), DU2( * ), DUF( * ),
309  \$ WORK( * ), X( LDX, * )
310 * ..
311 *
312 * =====================================================================
313 *
314 * .. Parameters ..
315  DOUBLE PRECISION ZERO
316  parameter( zero = 0.0d+0 )
317 * ..
318 * .. Local Scalars ..
319  LOGICAL NOFACT, NOTRAN
320  CHARACTER NORM
321  DOUBLE PRECISION ANORM
322 * ..
323 * .. External Functions ..
324  LOGICAL LSAME
325  DOUBLE PRECISION DLAMCH, ZLANGT
326  EXTERNAL lsame, dlamch, zlangt
327 * ..
328 * .. External Subroutines ..
329  EXTERNAL xerbla, zcopy, zgtcon, zgtrfs, zgttrf, zgttrs,
330  \$ zlacpy
331 * ..
332 * .. Intrinsic Functions ..
333  INTRINSIC max
334 * ..
335 * .. Executable Statements ..
336 *
337  info = 0
338  nofact = lsame( fact, 'N' )
339  notran = lsame( trans, 'N' )
340  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
341  info = -1
342  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
343  \$ lsame( trans, 'C' ) ) THEN
344  info = -2
345  ELSE IF( n.LT.0 ) THEN
346  info = -3
347  ELSE IF( nrhs.LT.0 ) THEN
348  info = -4
349  ELSE IF( ldb.LT.max( 1, n ) ) THEN
350  info = -14
351  ELSE IF( ldx.LT.max( 1, n ) ) THEN
352  info = -16
353  END IF
354  IF( info.NE.0 ) THEN
355  CALL xerbla( 'ZGTSVX', -info )
356  RETURN
357  END IF
358 *
359  IF( nofact ) THEN
360 *
361 * Compute the LU factorization of A.
362 *
363  CALL zcopy( n, d, 1, df, 1 )
364  IF( n.GT.1 ) THEN
365  CALL zcopy( n-1, dl, 1, dlf, 1 )
366  CALL zcopy( n-1, du, 1, duf, 1 )
367  END IF
368  CALL zgttrf( n, dlf, df, duf, du2, ipiv, info )
369 *
370 * Return if INFO is non-zero.
371 *
372  IF( info.GT.0 )THEN
373  rcond = zero
374  RETURN
375  END IF
376  END IF
377 *
378 * Compute the norm of the matrix A.
379 *
380  IF( notran ) THEN
381  norm = '1'
382  ELSE
383  norm = 'I'
384  END IF
385  anorm = zlangt( norm, n, dl, d, du )
386 *
387 * Compute the reciprocal of the condition number of A.
388 *
389  CALL zgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
390  \$ info )
391 *
392 * Compute the solution vectors X.
393 *
394  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395  CALL zgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
396  \$ info )
397 *
398 * Use iterative refinement to improve the computed solutions and
399 * compute error bounds and backward error estimates for them.
400 *
401  CALL zgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
402  \$ b, ldb, x, ldx, ferr, berr, work, rwork, info )
403 *
404 * Set INFO = N+1 if the matrix is singular to working precision.
405 *
406  IF( rcond.LT.dlamch( 'Epsilon' ) )
407  \$ info = n + 1
408 *
409  RETURN
410 *
411 * End of ZGTSVX
412 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgttrf(N, DL, D, DU, DU2, IPIV, INFO)
ZGTTRF
Definition: zgttrf.f:124
subroutine zgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)
ZGTCON
Definition: zgtcon.f:141
subroutine zgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
ZGTTRS
Definition: zgttrs.f:138
subroutine zgtrfs(TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZGTRFS
Definition: zgtrfs.f:210
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
double precision function zlangt(NORM, N, DL, D, DU)
ZLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlangt.f:106
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