LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sptsvx()

 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.```

Definition at line 226 of file sptsvx.f.

228 *
229 * -- LAPACK driver routine --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 *
233 * .. Scalar Arguments ..
234  CHARACTER FACT
235  INTEGER INFO, LDB, LDX, N, NRHS
236  REAL RCOND
237 * ..
238 * .. Array Arguments ..
239  REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
240  \$ E( * ), EF( * ), FERR( * ), WORK( * ),
241  \$ X( LDX, * )
242 * ..
243 *
244 * =====================================================================
245 *
246 * .. Parameters ..
247  REAL ZERO
248  parameter( zero = 0.0e+0 )
249 * ..
250 * .. Local Scalars ..
251  LOGICAL NOFACT
252  REAL ANORM
253 * ..
254 * .. External Functions ..
255  LOGICAL LSAME
256  REAL SLAMCH, SLANST
257  EXTERNAL lsame, slamch, slanst
258 * ..
259 * .. External Subroutines ..
260  EXTERNAL scopy, slacpy, sptcon, sptrfs, spttrf, spttrs,
261  \$ xerbla
262 * ..
263 * .. Intrinsic Functions ..
264  INTRINSIC max
265 * ..
266 * .. Executable Statements ..
267 *
268 * Test the input parameters.
269 *
270  info = 0
271  nofact = lsame( fact, 'N' )
272  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
273  info = -1
274  ELSE IF( n.LT.0 ) THEN
275  info = -2
276  ELSE IF( nrhs.LT.0 ) THEN
277  info = -3
278  ELSE IF( ldb.LT.max( 1, n ) ) THEN
279  info = -9
280  ELSE IF( ldx.LT.max( 1, n ) ) THEN
281  info = -11
282  END IF
283  IF( info.NE.0 ) THEN
284  CALL xerbla( 'SPTSVX', -info )
285  RETURN
286  END IF
287 *
288  IF( nofact ) THEN
289 *
290 * Compute the L*D*L**T (or U**T*D*U) factorization of A.
291 *
292  CALL scopy( n, d, 1, df, 1 )
293  IF( n.GT.1 )
294  \$ CALL scopy( n-1, e, 1, ef, 1 )
295  CALL spttrf( n, df, ef, info )
296 *
297 * Return if INFO is non-zero.
298 *
299  IF( info.GT.0 )THEN
300  rcond = zero
301  RETURN
302  END IF
303  END IF
304 *
305 * Compute the norm of the matrix A.
306 *
307  anorm = slanst( '1', n, d, e )
308 *
309 * Compute the reciprocal of the condition number of A.
310 *
311  CALL sptcon( n, df, ef, anorm, rcond, work, info )
312 *
313 * Compute the solution vectors X.
314 *
315  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
316  CALL spttrs( n, nrhs, df, ef, x, ldx, info )
317 *
318 * Use iterative refinement to improve the computed solutions and
319 * compute error bounds and backward error estimates for them.
320 *
321  CALL sptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr,
322  \$ work, info )
323 *
324 * Set INFO = N+1 if the matrix is singular to working precision.
325 *
326  IF( rcond.LT.slamch( 'Epsilon' ) )
327  \$ info = n + 1
328 *
329  RETURN
330 *
331 * End of SPTSVX
332 *
real function slanst(NORM, N, D, E)
SLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slanst.f:100
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine spttrf(N, D, E, INFO)
SPTTRF
Definition: spttrf.f:91
subroutine sptrfs(N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
SPTRFS
Definition: sptrfs.f:163
subroutine sptcon(N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON
Definition: sptcon.f:118
subroutine spttrs(N, NRHS, D, E, B, LDB, INFO)
SPTTRS
Definition: spttrs.f:109
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
Here is the call graph for this function:
Here is the caller graph for this function: