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

Functions

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

Detailed Description

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

Function Documentation

subroutine dptsv ( integer  N,
integer  NRHS,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldb, * )  B,
integer  LDB,
integer  INFO 
)

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

Download DPTSV + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DPTSV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
September 2012

Definition at line 116 of file dptsv.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  DOUBLE PRECISION b( ldb, * ), d( * ), e( * )
127 * ..
128 *
129 * =====================================================================
130 *
131 * .. External Subroutines ..
132  EXTERNAL dpttrf, dpttrs, 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( 'DPTSV ', -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 dpttrf( 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 dpttrs( n, nrhs, d, e, b, ldb, info )
162  END IF
163  RETURN
164 *
165 * End of DPTSV
166 *
subroutine dpttrf(N, D, E, INFO)
DPTTRF
Definition: dpttrf.f:93
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dpttrs(N, NRHS, D, E, B, LDB, INFO)
DPTTRS
Definition: dpttrs.f:111

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subroutine dptsvx ( character  FACT,
integer  N,
integer  NRHS,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( * )  DF,
double precision, dimension( * )  EF,
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  INFO 
)

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

Download DPTSVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DPTSVX 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 DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.
[in]E
          E is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix 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 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 DOUBLE PRECISION 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 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 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 DOUBLE PRECISION
          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 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 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 (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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
September 2012

Definition at line 230 of file dptsvx.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  DOUBLE PRECISION rcond
240 * ..
241 * .. Array Arguments ..
242  DOUBLE PRECISION b( ldb, * ), berr( * ), d( * ), df( * ),
243  $ e( * ), ef( * ), ferr( * ), work( * ),
244  $ x( ldx, * )
245 * ..
246 *
247 * =====================================================================
248 *
249 * .. Parameters ..
250  DOUBLE PRECISION zero
251  parameter( zero = 0.0d+0 )
252 * ..
253 * .. Local Scalars ..
254  LOGICAL nofact
255  DOUBLE PRECISION anorm
256 * ..
257 * .. External Functions ..
258  LOGICAL lsame
259  DOUBLE PRECISION dlamch, dlanst
260  EXTERNAL lsame, dlamch, dlanst
261 * ..
262 * .. External Subroutines ..
263  EXTERNAL dcopy, dlacpy, dptcon, dptrfs, dpttrf, dpttrs,
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( 'DPTSVX', -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 dcopy( n, d, 1, df, 1 )
296  IF( n.GT.1 )
297  $ CALL dcopy( n-1, e, 1, ef, 1 )
298  CALL dpttrf( 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 = dlanst( '1', n, d, e )
311 *
312 * Compute the reciprocal of the condition number of A.
313 *
314  CALL dptcon( n, df, ef, anorm, rcond, work, info )
315 *
316 * Compute the solution vectors X.
317 *
318  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
319  CALL dpttrs( 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 dptrfs( 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.dlamch( 'Epsilon' ) )
330  $ info = n + 1
331 *
332  RETURN
333 *
334 * End of DPTSVX
335 *
subroutine dpttrf(N, D, E, INFO)
DPTTRF
Definition: dpttrf.f:93
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 dptrfs(N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
DPTRFS
Definition: dptrfs.f:165
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlanst(NORM, N, D, E)
DLANST 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: dlanst.f:102
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dpttrs(N, NRHS, D, E, B, LDB, INFO)
DPTTRS
Definition: dpttrs.f:111
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dptcon(N, D, E, ANORM, RCOND, WORK, INFO)
DPTCON
Definition: dptcon.f:120
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53

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