LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
Collaboration diagram for double:


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## Functions

subroutine dgtsv (N, NRHS, DL, D, DU, B, LDB, INFO)
DGTSV computes the solution to system of linear equations A * X = B for GT matrices More...

subroutine dgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
DGTSVX computes the solution to system of linear equations A * X = B for GT matrices More...

## Detailed Description

This is the group of double solve driver functions for GT matrices

## Function Documentation

 subroutine dgtsv ( integer N, integer NRHS, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( ldb, * ) B, integer LDB, integer INFO )

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

Purpose:
``` DGTSV  solves the equation

A*X = B,

where A is an n by n tridiagonal matrix, by Gaussian elimination with
partial pivoting.

Note that the equation  A**T*X = B  may be solved by interchanging the
order of the arguments DU and DL.```
Parameters
 [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,out] DL ``` DL is DOUBLE PRECISION array, dimension (N-1) On entry, DL must contain the (n-1) sub-diagonal elements of A. On exit, DL is overwritten by the (n-2) elements of the second super-diagonal of the upper triangular matrix U from the LU factorization of A, in DL(1), ..., DL(n-2).``` [in,out] D ``` D is DOUBLE PRECISION array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of U.``` [in,out] DU ``` DU is DOUBLE PRECISION array, dimension (N-1) On entry, DU must contain the (n-1) super-diagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U.``` [in,out] B ``` B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N by NRHS matrix of right hand side matrix B. On exit, if INFO = 0, the N by NRHS solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,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, U(i,i) is exactly zero, and the solution has not been computed. The factorization has not been completed unless i = N.```
Date
September 2012

Definition at line 129 of file dgtsv.f.

129 *
130 * -- LAPACK driver routine (version 3.4.2) --
131 * -- LAPACK is a software package provided by Univ. of Tennessee, --
132 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133 * September 2012
134 *
135 * .. Scalar Arguments ..
136  INTEGER info, ldb, n, nrhs
137 * ..
138 * .. Array Arguments ..
139  DOUBLE PRECISION b( ldb, * ), d( * ), dl( * ), du( * )
140 * ..
141 *
142 * =====================================================================
143 *
144 * .. Parameters ..
145  DOUBLE PRECISION zero
146  parameter( zero = 0.0d+0 )
147 * ..
148 * .. Local Scalars ..
149  INTEGER i, j
150  DOUBLE PRECISION fact, temp
151 * ..
152 * .. Intrinsic Functions ..
153  INTRINSIC abs, max
154 * ..
155 * .. External Subroutines ..
156  EXTERNAL xerbla
157 * ..
158 * .. Executable Statements ..
159 *
160  info = 0
161  IF( n.LT.0 ) THEN
162  info = -1
163  ELSE IF( nrhs.LT.0 ) THEN
164  info = -2
165  ELSE IF( ldb.LT.max( 1, n ) ) THEN
166  info = -7
167  END IF
168  IF( info.NE.0 ) THEN
169  CALL xerbla( 'DGTSV ', -info )
170  RETURN
171  END IF
172 *
173  IF( n.EQ.0 )
174  \$ RETURN
175 *
176  IF( nrhs.EQ.1 ) THEN
177  DO 10 i = 1, n - 2
178  IF( abs( d( i ) ).GE.abs( dl( i ) ) ) THEN
179 *
180 * No row interchange required
181 *
182  IF( d( i ).NE.zero ) THEN
183  fact = dl( i ) / d( i )
184  d( i+1 ) = d( i+1 ) - fact*du( i )
185  b( i+1, 1 ) = b( i+1, 1 ) - fact*b( i, 1 )
186  ELSE
187  info = i
188  RETURN
189  END IF
190  dl( i ) = zero
191  ELSE
192 *
193 * Interchange rows I and I+1
194 *
195  fact = d( i ) / dl( i )
196  d( i ) = dl( i )
197  temp = d( i+1 )
198  d( i+1 ) = du( i ) - fact*temp
199  dl( i ) = du( i+1 )
200  du( i+1 ) = -fact*dl( i )
201  du( i ) = temp
202  temp = b( i, 1 )
203  b( i, 1 ) = b( i+1, 1 )
204  b( i+1, 1 ) = temp - fact*b( i+1, 1 )
205  END IF
206  10 CONTINUE
207  IF( n.GT.1 ) THEN
208  i = n - 1
209  IF( abs( d( i ) ).GE.abs( dl( i ) ) ) THEN
210  IF( d( i ).NE.zero ) THEN
211  fact = dl( i ) / d( i )
212  d( i+1 ) = d( i+1 ) - fact*du( i )
213  b( i+1, 1 ) = b( i+1, 1 ) - fact*b( i, 1 )
214  ELSE
215  info = i
216  RETURN
217  END IF
218  ELSE
219  fact = d( i ) / dl( i )
220  d( i ) = dl( i )
221  temp = d( i+1 )
222  d( i+1 ) = du( i ) - fact*temp
223  du( i ) = temp
224  temp = b( i, 1 )
225  b( i, 1 ) = b( i+1, 1 )
226  b( i+1, 1 ) = temp - fact*b( i+1, 1 )
227  END IF
228  END IF
229  IF( d( n ).EQ.zero ) THEN
230  info = n
231  RETURN
232  END IF
233  ELSE
234  DO 40 i = 1, n - 2
235  IF( abs( d( i ) ).GE.abs( dl( i ) ) ) THEN
236 *
237 * No row interchange required
238 *
239  IF( d( i ).NE.zero ) THEN
240  fact = dl( i ) / d( i )
241  d( i+1 ) = d( i+1 ) - fact*du( i )
242  DO 20 j = 1, nrhs
243  b( i+1, j ) = b( i+1, j ) - fact*b( i, j )
244  20 CONTINUE
245  ELSE
246  info = i
247  RETURN
248  END IF
249  dl( i ) = zero
250  ELSE
251 *
252 * Interchange rows I and I+1
253 *
254  fact = d( i ) / dl( i )
255  d( i ) = dl( i )
256  temp = d( i+1 )
257  d( i+1 ) = du( i ) - fact*temp
258  dl( i ) = du( i+1 )
259  du( i+1 ) = -fact*dl( i )
260  du( i ) = temp
261  DO 30 j = 1, nrhs
262  temp = b( i, j )
263  b( i, j ) = b( i+1, j )
264  b( i+1, j ) = temp - fact*b( i+1, j )
265  30 CONTINUE
266  END IF
267  40 CONTINUE
268  IF( n.GT.1 ) THEN
269  i = n - 1
270  IF( abs( d( i ) ).GE.abs( dl( i ) ) ) THEN
271  IF( d( i ).NE.zero ) THEN
272  fact = dl( i ) / d( i )
273  d( i+1 ) = d( i+1 ) - fact*du( i )
274  DO 50 j = 1, nrhs
275  b( i+1, j ) = b( i+1, j ) - fact*b( i, j )
276  50 CONTINUE
277  ELSE
278  info = i
279  RETURN
280  END IF
281  ELSE
282  fact = d( i ) / dl( i )
283  d( i ) = dl( i )
284  temp = d( i+1 )
285  d( i+1 ) = du( i ) - fact*temp
286  du( i ) = temp
287  DO 60 j = 1, nrhs
288  temp = b( i, j )
289  b( i, j ) = b( i+1, j )
290  b( i+1, j ) = temp - fact*b( i+1, j )
291  60 CONTINUE
292  END IF
293  END IF
294  IF( d( n ).EQ.zero ) THEN
295  info = n
296  RETURN
297  END IF
298  END IF
299 *
300 * Back solve with the matrix U from the factorization.
301 *
302  IF( nrhs.LE.2 ) THEN
303  j = 1
304  70 CONTINUE
305  b( n, j ) = b( n, j ) / d( n )
306  IF( n.GT.1 )
307  \$ b( n-1, j ) = ( b( n-1, j )-du( n-1 )*b( n, j ) ) / d( n-1 )
308  DO 80 i = n - 2, 1, -1
309  b( i, j ) = ( b( i, j )-du( i )*b( i+1, j )-dl( i )*
310  \$ b( i+2, j ) ) / d( i )
311  80 CONTINUE
312  IF( j.LT.nrhs ) THEN
313  j = j + 1
314  GO TO 70
315  END IF
316  ELSE
317  DO 100 j = 1, nrhs
318  b( n, j ) = b( n, j ) / d( n )
319  IF( n.GT.1 )
320  \$ b( n-1, j ) = ( b( n-1, j )-du( n-1 )*b( n, j ) ) /
321  \$ d( n-1 )
322  DO 90 i = n - 2, 1, -1
323  b( i, j ) = ( b( i, j )-du( i )*b( i+1, j )-dl( i )*
324  \$ b( i+2, j ) ) / d( i )
325  90 CONTINUE
326  100 CONTINUE
327  END IF
328 *
329  RETURN
330 *
331 * End of DGTSV
332 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62

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 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

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.```
Date
September 2012

Definition at line 295 of file dgtsvx.f.

295 *
296 * -- LAPACK driver routine (version 3.4.2) --
297 * -- LAPACK is a software package provided by Univ. of Tennessee, --
298 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
299 * September 2012
300 *
301 * .. Scalar Arguments ..
302  CHARACTER fact, trans
303  INTEGER info, ldb, ldx, n, nrhs
304  DOUBLE PRECISION rcond
305 * ..
306 * .. Array Arguments ..
307  INTEGER ipiv( * ), iwork( * )
308  DOUBLE PRECISION b( ldb, * ), berr( * ), d( * ), df( * ),
309  \$ dl( * ), dlf( * ), du( * ), du2( * ), duf( * ),
310  \$ ferr( * ), work( * ), x( ldx, * )
311 * ..
312 *
313 * =====================================================================
314 *
315 * .. Parameters ..
316  DOUBLE PRECISION zero
317  parameter( zero = 0.0d+0 )
318 * ..
319 * .. Local Scalars ..
320  LOGICAL nofact, notran
321  CHARACTER norm
322  DOUBLE PRECISION anorm
323 * ..
324 * .. External Functions ..
325  LOGICAL lsame
326  DOUBLE PRECISION dlamch, dlangt
327  EXTERNAL lsame, dlamch, dlangt
328 * ..
329 * .. External Subroutines ..
330  EXTERNAL dcopy, dgtcon, dgtrfs, dgttrf, dgttrs, dlacpy,
331  \$ xerbla
332 * ..
333 * .. Intrinsic Functions ..
334  INTRINSIC max
335 * ..
336 * .. Executable Statements ..
337 *
338  info = 0
339  nofact = lsame( fact, 'N' )
340  notran = lsame( trans, 'N' )
341  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
342  info = -1
343  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
344  \$ lsame( trans, 'C' ) ) THEN
345  info = -2
346  ELSE IF( n.LT.0 ) THEN
347  info = -3
348  ELSE IF( nrhs.LT.0 ) THEN
349  info = -4
350  ELSE IF( ldb.LT.max( 1, n ) ) THEN
351  info = -14
352  ELSE IF( ldx.LT.max( 1, n ) ) THEN
353  info = -16
354  END IF
355  IF( info.NE.0 ) THEN
356  CALL xerbla( 'DGTSVX', -info )
357  RETURN
358  END IF
359 *
360  IF( nofact ) THEN
361 *
362 * Compute the LU factorization of A.
363 *
364  CALL dcopy( n, d, 1, df, 1 )
365  IF( n.GT.1 ) THEN
366  CALL dcopy( n-1, dl, 1, dlf, 1 )
367  CALL dcopy( n-1, du, 1, duf, 1 )
368  END IF
369  CALL dgttrf( n, dlf, df, duf, du2, ipiv, info )
370 *
371 * Return if INFO is non-zero.
372 *
373  IF( info.GT.0 )THEN
374  rcond = zero
375  RETURN
376  END IF
377  END IF
378 *
379 * Compute the norm of the matrix A.
380 *
381  IF( notran ) THEN
382  norm = '1'
383  ELSE
384  norm = 'I'
385  END IF
386  anorm = dlangt( norm, n, dl, d, du )
387 *
388 * Compute the reciprocal of the condition number of A.
389 *
390  CALL dgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
391  \$ iwork, info )
392 *
393 * Compute the solution vectors X.
394 *
395  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
396  CALL dgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
397  \$ info )
398 *
399 * Use iterative refinement to improve the computed solutions and
400 * compute error bounds and backward error estimates for them.
401 *
402  CALL dgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
403  \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
404 *
405 * Set INFO = N+1 if the matrix is singular to working precision.
406 *
407  IF( rcond.LT.dlamch( 'Epsilon' ) )
408  \$ info = n + 1
409 *
410  RETURN
411 *
412 * End of DGTSVX
413 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dgttrf(N, DL, D, DU, DU2, IPIV, INFO)
DGTTRF
Definition: dgttrf.f:126
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DGTCON
Definition: dgtcon.f:148
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:108
subroutine dgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
DGTTRS
Definition: dgttrs.f:140
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:211
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53

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