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

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

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

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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