LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
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## Functions

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

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

## Detailed Description

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

## Function Documentation

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

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

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

A is factored as A = L*D*L**T, 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**T.``` [in,out] E ``` E is REAL 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**T factorization of A. (E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**T*D*U factorization of A.)``` [in,out] B ``` B is REAL 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 116 of file sptsv.f.

116 *
117 * -- LAPACK driver routine (version 3.4.2) --
118 * -- LAPACK is a software package provided by Univ. of Tennessee, --
119 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120 * September 2012
121 *
122 * .. Scalar Arguments ..
123  INTEGER info, ldb, n, nrhs
124 * ..
125 * .. Array Arguments ..
126  REAL b( ldb, * ), d( * ), e( * )
127 * ..
128 *
129 * =====================================================================
130 *
131 * .. External Subroutines ..
132  EXTERNAL spttrf, spttrs, xerbla
133 * ..
134 * .. Intrinsic Functions ..
135  INTRINSIC max
136 * ..
137 * .. Executable Statements ..
138 *
139 * Test the input parameters.
140 *
141  info = 0
142  IF( n.LT.0 ) THEN
143  info = -1
144  ELSE IF( nrhs.LT.0 ) THEN
145  info = -2
146  ELSE IF( ldb.LT.max( 1, n ) ) THEN
147  info = -6
148  END IF
149  IF( info.NE.0 ) THEN
150  CALL xerbla( 'SPTSV ', -info )
151  RETURN
152  END IF
153 *
154 * Compute the L*D*L**T (or U**T*D*U) factorization of A.
155 *
156  CALL spttrf( n, d, e, info )
157  IF( info.EQ.0 ) THEN
158 *
159 * Solve the system A*X = B, overwriting B with X.
160 *
161  CALL spttrs( n, nrhs, d, e, b, ldb, info )
162  END IF
163  RETURN
164 *
165 * End of SPTSV
166 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine spttrf(N, D, E, INFO)
SPTTRF
Definition: spttrf.f:93
subroutine spttrs(N, NRHS, D, E, B, LDB, INFO)
SPTTRS
Definition: spttrs.f:111

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 subroutine sptsvx ( character FACT, integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO )

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

Purpose:
``` SPTSVX uses the factorization A = L*D*L**T to compute the solution
to a real system of linear equations A*X = B, where A is an N-by-N
symmetric 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**T, where L
is a unit lower bidiagonal matrix and D is diagonal.  The
factorization can also be regarded as having the form
A = U**T*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 A has been 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 REAL 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**T 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**T factorization of A.``` [in,out] EF ``` EF is REAL 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**T 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**T factorization of A.``` [in] B ``` B is REAL 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 REAL array, dimension (LDX,NRHS) If INFO = 0 of 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 REAL array, dimension (2*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 230 of file sptsvx.f.

230 *
231 * -- LAPACK driver routine (version 3.4.2) --
232 * -- LAPACK is a software package provided by Univ. of Tennessee, --
233 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
234 * September 2012
235 *
236 * .. Scalar Arguments ..
237  CHARACTER fact
238  INTEGER info, ldb, ldx, n, nrhs
239  REAL rcond
240 * ..
241 * .. Array Arguments ..
242  REAL b( ldb, * ), berr( * ), d( * ), df( * ),
243  \$ e( * ), ef( * ), ferr( * ), work( * ),
244  \$ x( ldx, * )
245 * ..
246 *
247 * =====================================================================
248 *
249 * .. Parameters ..
250  REAL zero
251  parameter( zero = 0.0e+0 )
252 * ..
253 * .. Local Scalars ..
254  LOGICAL nofact
255  REAL anorm
256 * ..
257 * .. External Functions ..
258  LOGICAL lsame
259  REAL slamch, slanst
260  EXTERNAL lsame, slamch, slanst
261 * ..
262 * .. External Subroutines ..
263  EXTERNAL scopy, slacpy, sptcon, sptrfs, spttrf, spttrs,
264  \$ xerbla
265 * ..
266 * .. Intrinsic Functions ..
267  INTRINSIC max
268 * ..
269 * .. Executable Statements ..
270 *
271 * Test the input parameters.
272 *
273  info = 0
274  nofact = lsame( fact, 'N' )
275  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
276  info = -1
277  ELSE IF( n.LT.0 ) THEN
278  info = -2
279  ELSE IF( nrhs.LT.0 ) THEN
280  info = -3
281  ELSE IF( ldb.LT.max( 1, n ) ) THEN
282  info = -9
283  ELSE IF( ldx.LT.max( 1, n ) ) THEN
284  info = -11
285  END IF
286  IF( info.NE.0 ) THEN
287  CALL xerbla( 'SPTSVX', -info )
288  RETURN
289  END IF
290 *
291  IF( nofact ) THEN
292 *
293 * Compute the L*D*L**T (or U**T*D*U) factorization of A.
294 *
295  CALL scopy( n, d, 1, df, 1 )
296  IF( n.GT.1 )
297  \$ CALL scopy( n-1, e, 1, ef, 1 )
298  CALL spttrf( n, df, ef, info )
299 *
300 * Return if INFO is non-zero.
301 *
302  IF( info.GT.0 )THEN
303  rcond = zero
304  RETURN
305  END IF
306  END IF
307 *
308 * Compute the norm of the matrix A.
309 *
310  anorm = slanst( '1', n, d, e )
311 *
312 * Compute the reciprocal of the condition number of A.
313 *
314  CALL sptcon( n, df, ef, anorm, rcond, work, info )
315 *
316 * Compute the solution vectors X.
317 *
318  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
319  CALL spttrs( n, nrhs, df, ef, x, ldx, info )
320 *
321 * Use iterative refinement to improve the computed solutions and
322 * compute error bounds and backward error estimates for them.
323 *
324  CALL sptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr,
325  \$ work, info )
326 *
327 * Set INFO = N+1 if the matrix is singular to working precision.
328 *
329  IF( rcond.LT.slamch( 'Epsilon' ) )
330  \$ info = n + 1
331 *
332  RETURN
333 *
334 * End of SPTSVX
335 *
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 sptrfs(N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
SPTRFS
Definition: sptrfs.f:165
real function slanst(NORM, N, D, E)
SLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
Definition: slanst.f:102
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine sptcon(N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON
Definition: sptcon.f:120
subroutine spttrf(N, D, E, INFO)
SPTTRF
Definition: spttrf.f:93
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine spttrs(N, NRHS, D, E, B, LDB, INFO)
SPTTRS
Definition: spttrs.f:111

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