LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ dgtsvx()

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

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

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

Purpose:
 DGTSVX 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 DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of A.
[in]D
          D is DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of A.
[in]DU
          DU is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) superdiagonal elements of A.
[in,out]DLF
          DLF is DOUBLE PRECISION 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 DGTTRF.

          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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGTTRF.

          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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 290 of file dgtsvx.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  DOUBLE PRECISION RCOND
302 * ..
303 * .. Array Arguments ..
304  INTEGER IPIV( * ), IWORK( * )
305  DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
306  $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
307  $ FERR( * ), WORK( * ), X( LDX, * )
308 * ..
309 *
310 * =====================================================================
311 *
312 * .. Parameters ..
313  DOUBLE PRECISION ZERO
314  parameter( zero = 0.0d+0 )
315 * ..
316 * .. Local Scalars ..
317  LOGICAL NOFACT, NOTRAN
318  CHARACTER NORM
319  DOUBLE PRECISION ANORM
320 * ..
321 * .. External Functions ..
322  LOGICAL LSAME
323  DOUBLE PRECISION DLAMCH, DLANGT
324  EXTERNAL lsame, dlamch, dlangt
325 * ..
326 * .. External Subroutines ..
327  EXTERNAL dcopy, dgtcon, dgtrfs, dgttrf, dgttrs, dlacpy,
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( 'DGTSVX', -info )
354  RETURN
355  END IF
356 *
357  IF( nofact ) THEN
358 *
359 * Compute the LU factorization of A.
360 *
361  CALL dcopy( n, d, 1, df, 1 )
362  IF( n.GT.1 ) THEN
363  CALL dcopy( n-1, dl, 1, dlf, 1 )
364  CALL dcopy( n-1, du, 1, duf, 1 )
365  END IF
366  CALL dgttrf( 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 = dlangt( norm, n, dl, d, du )
384 *
385 * Compute the reciprocal of the condition number of A.
386 *
387  CALL dgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
388  $ iwork, info )
389 *
390 * Compute the solution vectors X.
391 *
392  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
393  CALL dgttrs( 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 dgtrfs( 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.dlamch( 'Epsilon' ) )
405  $ info = n + 1
406 *
407  RETURN
408 *
409 * End of DGTSVX
410 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dgtrfs(TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGTRFS
Definition: dgtrfs.f:209
subroutine dgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DGTCON
Definition: dgtcon.f:146
subroutine dgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
DGTTRS
Definition: dgttrs.f:138
subroutine dgttrf(N, DL, D, DU, DU2, IPIV, INFO)
DGTTRF
Definition: dgttrf.f:124
double precision function dlangt(NORM, N, DL, D, DU)
DLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlangt.f:106
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