LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
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## Functions

subroutine cptsv (N, NRHS, D, E, B, LDB, INFO)
CPTSV computes the solution to system of linear equations A * X = B for PT matrices More...

subroutine cptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CPTSVX computes the solution to system of linear equations A * X = B for PT matrices More...

## Detailed Description

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

## Function Documentation

 subroutine cptsv ( integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB, integer INFO )

CPTSV computes the solution to system of linear equations A * X = B for PT matrices

Purpose:
``` CPTSV computes the solution to a complex system of linear equations
A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal
matrix, and X and B are N-by-NRHS matrices.

A is factored as A = L*D*L**H, and the factored form of A is then
used to solve the system of equations.```
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] D ``` D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the factorization A = L*D*L**H.``` [in,out] E ``` E is COMPLEX array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**H*D*U factorization of A.``` [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, the leading minor of order i is not positive definite, and the solution has not been computed. The factorization has not been completed unless i = N.```
Date
September 2012

Definition at line 117 of file cptsv.f.

117 *
118 * -- LAPACK driver routine (version 3.4.2) --
119 * -- LAPACK is a software package provided by Univ. of Tennessee, --
120 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
121 * September 2012
122 *
123 * .. Scalar Arguments ..
124  INTEGER info, ldb, n, nrhs
125 * ..
126 * .. Array Arguments ..
127  REAL d( * )
128  COMPLEX b( ldb, * ), e( * )
129 * ..
130 *
131 * =====================================================================
132 *
133 * .. External Subroutines ..
134  EXTERNAL cpttrf, cpttrs, xerbla
135 * ..
136 * .. Intrinsic Functions ..
137  INTRINSIC max
138 * ..
139 * .. Executable Statements ..
140 *
141 * Test the input parameters.
142 *
143  info = 0
144  IF( n.LT.0 ) THEN
145  info = -1
146  ELSE IF( nrhs.LT.0 ) THEN
147  info = -2
148  ELSE IF( ldb.LT.max( 1, n ) ) THEN
149  info = -6
150  END IF
151  IF( info.NE.0 ) THEN
152  CALL xerbla( 'CPTSV ', -info )
153  RETURN
154  END IF
155 *
156 * Compute the L*D*L**H (or U**H*D*U) factorization of A.
157 *
158  CALL cpttrf( n, d, e, info )
159  IF( info.EQ.0 ) THEN
160 *
161 * Solve the system A*X = B, overwriting B with X.
162 *
163  CALL cpttrs( 'Lower', n, nrhs, d, e, b, ldb, info )
164  END IF
165  RETURN
166 *
167 * End of CPTSV
168 *
subroutine cpttrs(UPLO, N, NRHS, D, E, B, LDB, INFO)
CPTTRS
Definition: cpttrs.f:123
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cpttrf(N, D, E, INFO)
CPTTRF
Definition: cpttrf.f:94

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 subroutine cptsvx ( character FACT, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, real, dimension( * ) DF, complex, dimension( * ) EF, 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 )

CPTSVX computes the solution to system of linear equations A * X = B for PT matrices

Purpose:
``` CPTSVX uses the factorization A = L*D*L**H to compute the solution
to a complex system of linear equations A*X = B, where A is an
N-by-N Hermitian positive definite tridiagonal matrix 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 matrix A is factored as A = L*D*L**H, where L
is a unit lower bidiagonal matrix and D is diagonal.  The
factorization can also be regarded as having the form
A = U**H*D*U.

2. If the leading i-by-i principal minor is not positive definite,
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 the matrix A is supplied on entry. = 'F': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = 'N': The matrix A will be copied to DF and EF and factored.``` [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 matrices B and X. NRHS >= 0.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix A.``` [in] E ``` E is COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix A.``` [in,out] DF ``` DF is REAL array, dimension (N) If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A.``` [in,out] EF ``` EF is COMPLEX array, dimension (N-1) If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A.``` [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 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 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).``` [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 (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: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been 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 236 of file cptsvx.f.

236 *
237 * -- LAPACK driver routine (version 3.4.2) --
238 * -- LAPACK is a software package provided by Univ. of Tennessee, --
239 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
240 * September 2012
241 *
242 * .. Scalar Arguments ..
243  CHARACTER fact
244  INTEGER info, ldb, ldx, n, nrhs
245  REAL rcond
246 * ..
247 * .. Array Arguments ..
248  REAL berr( * ), d( * ), df( * ), ferr( * ),
249  \$ rwork( * )
250  COMPLEX b( ldb, * ), e( * ), ef( * ), work( * ),
251  \$ x( ldx, * )
252 * ..
253 *
254 * =====================================================================
255 *
256 * .. Parameters ..
257  REAL zero
258  parameter( zero = 0.0e+0 )
259 * ..
260 * .. Local Scalars ..
261  LOGICAL nofact
262  REAL anorm
263 * ..
264 * .. External Functions ..
265  LOGICAL lsame
266  REAL clanht, slamch
267  EXTERNAL lsame, clanht, slamch
268 * ..
269 * .. External Subroutines ..
270  EXTERNAL ccopy, clacpy, cptcon, cptrfs, cpttrf, cpttrs,
271  \$ scopy, xerbla
272 * ..
273 * .. Intrinsic Functions ..
274  INTRINSIC max
275 * ..
276 * .. Executable Statements ..
277 *
278 * Test the input parameters.
279 *
280  info = 0
281  nofact = lsame( fact, 'N' )
282  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
283  info = -1
284  ELSE IF( n.LT.0 ) THEN
285  info = -2
286  ELSE IF( nrhs.LT.0 ) THEN
287  info = -3
288  ELSE IF( ldb.LT.max( 1, n ) ) THEN
289  info = -9
290  ELSE IF( ldx.LT.max( 1, n ) ) THEN
291  info = -11
292  END IF
293  IF( info.NE.0 ) THEN
294  CALL xerbla( 'CPTSVX', -info )
295  RETURN
296  END IF
297 *
298  IF( nofact ) THEN
299 *
300 * Compute the L*D*L**H (or U**H*D*U) factorization of A.
301 *
302  CALL scopy( n, d, 1, df, 1 )
303  IF( n.GT.1 )
304  \$ CALL ccopy( n-1, e, 1, ef, 1 )
305  CALL cpttrf( n, df, ef, info )
306 *
307 * Return if INFO is non-zero.
308 *
309  IF( info.GT.0 )THEN
310  rcond = zero
311  RETURN
312  END IF
313  END IF
314 *
315 * Compute the norm of the matrix A.
316 *
317  anorm = clanht( '1', n, d, e )
318 *
319 * Compute the reciprocal of the condition number of A.
320 *
321  CALL cptcon( n, df, ef, anorm, rcond, rwork, info )
322 *
323 * Compute the solution vectors X.
324 *
325  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
326  CALL cpttrs( 'Lower', n, nrhs, df, ef, x, ldx, info )
327 *
328 * Use iterative refinement to improve the computed solutions and
329 * compute error bounds and backward error estimates for them.
330 *
331  CALL cptrfs( 'Lower', n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr,
332  \$ berr, work, rwork, info )
333 *
334 * Set INFO = N+1 if the matrix is singular to working precision.
335 *
336  IF( rcond.LT.slamch( 'Epsilon' ) )
337  \$ info = n + 1
338 *
339  RETURN
340 *
341 * End of CPTSVX
342 *
subroutine cpttrs(UPLO, N, NRHS, D, E, B, LDB, INFO)
CPTTRS
Definition: cpttrs.f:123
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cpttrf(N, D, E, INFO)
CPTTRF
Definition: cpttrf.f:94
real function clanht(NORM, N, D, E)
CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix.
Definition: clanht.f:103
subroutine cptrfs(UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPTRFS
Definition: cptrfs.f:185
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 cptcon(N, D, E, ANORM, RCOND, RWORK, INFO)
CPTCON
Definition: cptcon.f:121
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:52

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