LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
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## Functions

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

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

## Detailed Description

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

## Function Documentation

 subroutine cgtsv ( integer N, integer NRHS, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( ldb, * ) B, integer LDB, integer INFO )

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

Purpose:
``` CGTSV  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 COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-2) elements of the second superdiagonal of the upper triangular matrix U from the LU factorization of A, in DL(1), ..., DL(n-2).``` [in,out] D ``` D is COMPLEX 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 COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U.``` [in,out] B ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS 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 126 of file cgtsv.f.

126 *
127 * -- LAPACK driver routine (version 3.4.2) --
128 * -- LAPACK is a software package provided by Univ. of Tennessee, --
129 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
130 * September 2012
131 *
132 * .. Scalar Arguments ..
133  INTEGER info, ldb, n, nrhs
134 * ..
135 * .. Array Arguments ..
136  COMPLEX b( ldb, * ), d( * ), dl( * ), du( * )
137 * ..
138 *
139 * =====================================================================
140 *
141 * .. Parameters ..
142  COMPLEX zero
143  parameter( zero = ( 0.0e+0, 0.0e+0 ) )
144 * ..
145 * .. Local Scalars ..
146  INTEGER j, k
147  COMPLEX mult, temp, zdum
148 * ..
149 * .. Intrinsic Functions ..
150  INTRINSIC abs, aimag, max, real
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL xerbla
154 * ..
155 * .. Statement Functions ..
156  REAL cabs1
157 * ..
158 * .. Statement Function definitions ..
159  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
160 * ..
161 * .. Executable Statements ..
162 *
163  info = 0
164  IF( n.LT.0 ) THEN
165  info = -1
166  ELSE IF( nrhs.LT.0 ) THEN
167  info = -2
168  ELSE IF( ldb.LT.max( 1, n ) ) THEN
169  info = -7
170  END IF
171  IF( info.NE.0 ) THEN
172  CALL xerbla( 'CGTSV ', -info )
173  RETURN
174  END IF
175 *
176  IF( n.EQ.0 )
177  \$ RETURN
178 *
179  DO 30 k = 1, n - 1
180  IF( dl( k ).EQ.zero ) THEN
181 *
182 * Subdiagonal is zero, no elimination is required.
183 *
184  IF( d( k ).EQ.zero ) THEN
185 *
186 * Diagonal is zero: set INFO = K and return; a unique
187 * solution can not be found.
188 *
189  info = k
190  RETURN
191  END IF
192  ELSE IF( cabs1( d( k ) ).GE.cabs1( dl( k ) ) ) THEN
193 *
194 * No row interchange required
195 *
196  mult = dl( k ) / d( k )
197  d( k+1 ) = d( k+1 ) - mult*du( k )
198  DO 10 j = 1, nrhs
199  b( k+1, j ) = b( k+1, j ) - mult*b( k, j )
200  10 CONTINUE
201  IF( k.LT.( n-1 ) )
202  \$ dl( k ) = zero
203  ELSE
204 *
205 * Interchange rows K and K+1
206 *
207  mult = d( k ) / dl( k )
208  d( k ) = dl( k )
209  temp = d( k+1 )
210  d( k+1 ) = du( k ) - mult*temp
211  IF( k.LT.( n-1 ) ) THEN
212  dl( k ) = du( k+1 )
213  du( k+1 ) = -mult*dl( k )
214  END IF
215  du( k ) = temp
216  DO 20 j = 1, nrhs
217  temp = b( k, j )
218  b( k, j ) = b( k+1, j )
219  b( k+1, j ) = temp - mult*b( k+1, j )
220  20 CONTINUE
221  END IF
222  30 CONTINUE
223  IF( d( n ).EQ.zero ) THEN
224  info = n
225  RETURN
226  END IF
227 *
228 * Back solve with the matrix U from the factorization.
229 *
230  DO 50 j = 1, nrhs
231  b( n, j ) = b( n, j ) / d( n )
232  IF( n.GT.1 )
233  \$ b( n-1, j ) = ( b( n-1, j )-du( n-1 )*b( n, j ) ) / d( n-1 )
234  DO 40 k = n - 2, 1, -1
235  b( k, j ) = ( b( k, j )-du( k )*b( k+1, j )-dl( k )*
236  \$ b( k+2, j ) ) / d( k )
237  40 CONTINUE
238  50 CONTINUE
239 *
240  RETURN
241 *
242 * End of CGTSV
243 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62

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 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

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 cgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)
CGTCON
Definition: cgtcon.f:143
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
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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 cgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
CGTTRS
Definition: cgttrs.f:140
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:52
subroutine cgttrf(N, DL, D, DU, DU2, IPIV, INFO)
CGTTRF
Definition: cgttrf.f:126

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