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authorjulie <julielangou@users.noreply.github.com>2008-12-16 17:06:58 +0000
committerjulie <julielangou@users.noreply.github.com>2008-12-16 17:06:58 +0000
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+ SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
+ $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
+ $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
+ $ NPARAMS, PARAMS, WORK, RWORK, INFO )
+*
+* -- LAPACK driver routine (version 3.2) --
+* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
+* -- Jason Riedy of Univ. of California Berkeley. --
+* -- November 2008 --
+*
+* -- LAPACK is a software package provided by Univ. of Tennessee, --
+* -- Univ. of California Berkeley and NAG Ltd. --
+*
+ IMPLICIT NONE
+* ..
+* .. Scalar Arguments ..
+ CHARACTER EQUED, FACT, UPLO
+ INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
+ $ N_ERR_BNDS
+ DOUBLE PRECISION RCOND, RPVGRW
+* ..
+* .. Array Arguments ..
+ COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
+ $ WORK( * ), X( LDX, * )
+ DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
+ $ ERR_BNDS_NORM( NRHS, * ),
+ $ ERR_BNDS_COMP( NRHS, * )
+* ..
+*
+* Purpose
+* =======
+*
+* ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
+* to compute the solution to a complex*16 system of linear equations
+* A * X = B, where A is an N-by-N symmetric positive definite matrix
+* and X and B are N-by-NRHS matrices.
+*
+* If requested, both normwise and maximum componentwise error bounds
+* are returned. ZPOSVXX will return a solution with a tiny
+* guaranteed error (O(eps) where eps is the working machine
+* precision) unless the matrix is very ill-conditioned, in which
+* case a warning is returned. Relevant condition numbers also are
+* calculated and returned.
+*
+* ZPOSVXX accepts user-provided factorizations and equilibration
+* factors; see the definitions of the FACT and EQUED options.
+* Solving with refinement and using a factorization from a previous
+* ZPOSVXX call will also produce a solution with either O(eps)
+* errors or warnings, but we cannot make that claim for general
+* user-provided factorizations and equilibration factors if they
+* differ from what ZPOSVXX would itself produce.
+*
+* Description
+* ===========
+*
+* The following steps are performed:
+*
+* 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
+* the system:
+*
+* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
+*
+* Whether or not the system will be equilibrated depends on the
+* scaling of the matrix A, but if equilibration is used, A is
+* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
+*
+* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
+* factor the matrix A (after equilibration if FACT = 'E') as
+* A = U**T* U, if UPLO = 'U', or
+* A = L * L**T, if UPLO = 'L',
+* where U is an upper triangular matrix and L is a lower triangular
+* matrix.
+*
+* 3. 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 (see argument RCOND). If the reciprocal of the condition number
+* is less than machine precision, the routine still goes on to solve
+* for X and compute error bounds as described below.
+*
+* 4. The system of equations is solved for X using the factored form
+* of A.
+*
+* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
+* the routine will use iterative refinement to try to get a small
+* error and error bounds. Refinement calculates the residual to at
+* least twice the working precision.
+*
+* 6. If equilibration was used, the matrix X is premultiplied by
+* diag(S) so that it solves the original system before
+* equilibration.
+*
+* Arguments
+* =========
+*
+* Some optional parameters are bundled in the PARAMS array. These
+* settings determine how refinement is performed, but often the
+* defaults are acceptable. If the defaults are acceptable, users
+* can pass NPARAMS = 0 which prevents the source code from accessing
+* the PARAMS argument.
+*
+* FACT (input) CHARACTER*1
+* Specifies whether or not the factored form of the matrix A is
+* supplied on entry, and if not, whether the matrix A should be
+* equilibrated before it is factored.
+* = 'F': On entry, AF contains the factored form of A.
+* If EQUED is not 'N', the matrix A has been
+* equilibrated with scaling factors given by S.
+* A and AF are not modified.
+* = 'N': The matrix A will be copied to AF and factored.
+* = 'E': The matrix A will be equilibrated if necessary, then
+* copied to AF and factored.
+*
+* UPLO (input) CHARACTER*1
+* = 'U': Upper triangle of A is stored;
+* = 'L': Lower triangle of A is stored.
+*
+* N (input) INTEGER
+* The number of linear equations, i.e., the order of the
+* matrix A. N >= 0.
+*
+* NRHS (input) INTEGER
+* The number of right hand sides, i.e., the number of columns
+* of the matrices B and X. NRHS >= 0.
+*
+* A (input/output) COMPLEX*16 array, dimension (LDA,N)
+* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
+* 'Y', then A must contain the equilibrated matrix
+* diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
+* triangular part of A contains the upper triangular part of the
+* matrix A, and the strictly lower triangular part of A is not
+* referenced. If UPLO = 'L', the leading N-by-N lower triangular
+* part of A contains the lower triangular part of the matrix A, and
+* the strictly upper triangular part of A is not referenced. A is
+* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
+* 'N' on exit.
+*
+* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
+* diag(S)*A*diag(S).
+*
+* LDA (input) INTEGER
+* The leading dimension of the array A. LDA >= max(1,N).
+*
+* AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
+* If FACT = 'F', then AF is an input argument and on entry
+* contains the triangular factor U or L from the Cholesky
+* factorization A = U**T*U or A = L*L**T, in the same storage
+* format as A. If EQUED .ne. 'N', then AF is the factored
+* form of the equilibrated matrix diag(S)*A*diag(S).
+*
+* If FACT = 'N', then AF is an output argument and on exit
+* returns the triangular factor U or L from the Cholesky
+* factorization A = U**T*U or A = L*L**T of the original
+* matrix A.
+*
+* If FACT = 'E', then AF is an output argument and on exit
+* returns the triangular factor U or L from the Cholesky
+* factorization A = U**T*U or A = L*L**T of the equilibrated
+* matrix A (see the description of A for the form of the
+* equilibrated matrix).
+*
+* LDAF (input) INTEGER
+* The leading dimension of the array AF. LDAF >= max(1,N).
+*
+* EQUED (input or output) CHARACTER*1
+* Specifies the form of equilibration that was done.
+* = 'N': No equilibration (always true if FACT = 'N').
+* = 'Y': Both row and column equilibration, i.e., A has been
+* replaced by diag(S) * A * diag(S).
+* EQUED is an input argument if FACT = 'F'; otherwise, it is an
+* output argument.
+*
+* S (input or output) DOUBLE PRECISION array, dimension (N)
+* The row scale factors for A. If EQUED = 'Y', A is multiplied on
+* the left and right by diag(S). S is an input argument if FACT =
+* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
+* = 'Y', each element of S must be positive. If S is output, each
+* element of S is a power of the radix. If S is input, each element
+* of S should be a power of the radix to ensure a reliable solution
+* and error estimates. Scaling by powers of the radix does not cause
+* rounding errors unless the result underflows or overflows.
+* Rounding errors during scaling lead to refining with a matrix that
+* is not equivalent to the input matrix, producing error estimates
+* that may not be reliable.
+*
+* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
+* On entry, the N-by-NRHS right hand side matrix B.
+* On exit,
+* if EQUED = 'N', B is not modified;
+* if EQUED = 'Y', B is overwritten by diag(S)*B;
+*
+* LDB (input) INTEGER
+* The leading dimension of the array B. LDB >= max(1,N).
+*
+* X (output) COMPLEX*16 array, dimension (LDX,NRHS)
+* If INFO = 0, the N-by-NRHS solution matrix X to the original
+* system of equations. Note that A and B are modified on exit if
+* EQUED .ne. 'N', and the solution to the equilibrated system is
+* inv(diag(S))*X.
+*
+* LDX (input) INTEGER
+* The leading dimension of the array X. LDX >= max(1,N).
+*
+* RCOND (output) DOUBLE PRECISION
+* Reciprocal scaled condition number. This is an estimate of the
+* reciprocal Skeel condition number of the matrix A after
+* equilibration (if done). If this is less than the machine
+* precision (in particular, if it is zero), the matrix is singular
+* to working precision. Note that the error may still be small even
+* if this number is very small and the matrix appears ill-
+* conditioned.
+*
+* RPVGRW (output) DOUBLE PRECISION
+* Reciprocal pivot growth. On exit, this contains the reciprocal
+* pivot growth factor norm(A)/norm(U). The "max absolute element"
+* norm is used. If this is much less than 1, then the stability of
+* the LU factorization of the (equilibrated) matrix A could be poor.
+* This also means that the solution X, estimated condition numbers,
+* and error bounds could be unreliable. If factorization fails with
+* 0<INFO<=N, then this contains the reciprocal pivot growth factor
+* for the leading INFO columns of A.
+*
+* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
+* Componentwise relative backward error. This is 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).
+*
+* N_ERR_BNDS (input) INTEGER
+* Number of error bounds to return for each right hand side
+* and each type (normwise or componentwise). See ERR_BNDS_NORM and
+* ERR_BNDS_COMP below.
+*
+* ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
+* For each right-hand side, this array contains information about
+* various error bounds and condition numbers corresponding to the
+* normwise relative error, which is defined as follows:
+*
+* Normwise relative error in the ith solution vector:
+* max_j (abs(XTRUE(j,i) - X(j,i)))
+* ------------------------------
+* max_j abs(X(j,i))
+*
+* The array is indexed by the type of error information as described
+* below. There currently are up to three pieces of information
+* returned.
+*
+* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
+* right-hand side.
+*
+* The second index in ERR_BNDS_NORM(:,err) contains the following
+* three fields:
+* err = 1 "Trust/don't trust" boolean. Trust the answer if the
+* reciprocal condition number is less than the threshold
+* sqrt(n) * dlamch('Epsilon').
+*
+* err = 2 "Guaranteed" error bound: The estimated forward error,
+* almost certainly within a factor of 10 of the true error
+* so long as the next entry is greater than the threshold
+* sqrt(n) * dlamch('Epsilon'). This error bound should only
+* be trusted if the previous boolean is true.
+*
+* err = 3 Reciprocal condition number: Estimated normwise
+* reciprocal condition number. Compared with the threshold
+* sqrt(n) * dlamch('Epsilon') to determine if the error
+* estimate is "guaranteed". These reciprocal condition
+* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
+* appropriately scaled matrix Z.
+* Let Z = S*A, where S scales each row by a power of the
+* radix so all absolute row sums of Z are approximately 1.
+*
+* See Lapack Working Note 165 for further details and extra
+* cautions.
+*
+* ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
+* For each right-hand side, this array contains information about
+* various error bounds and condition numbers corresponding to the
+* componentwise relative error, which is defined as follows:
+*
+* Componentwise relative error in the ith solution vector:
+* abs(XTRUE(j,i) - X(j,i))
+* max_j ----------------------
+* abs(X(j,i))
+*
+* The array is indexed by the right-hand side i (on which the
+* componentwise relative error depends), and the type of error
+* information as described below. There currently are up to three
+* pieces of information returned for each right-hand side. If
+* componentwise accuracy is not requested (PARAMS(3) = 0.0), then
+* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
+* the first (:,N_ERR_BNDS) entries are returned.
+*
+* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
+* right-hand side.
+*
+* The second index in ERR_BNDS_COMP(:,err) contains the following
+* three fields:
+* err = 1 "Trust/don't trust" boolean. Trust the answer if the
+* reciprocal condition number is less than the threshold
+* sqrt(n) * dlamch('Epsilon').
+*
+* err = 2 "Guaranteed" error bound: The estimated forward error,
+* almost certainly within a factor of 10 of the true error
+* so long as the next entry is greater than the threshold
+* sqrt(n) * dlamch('Epsilon'). This error bound should only
+* be trusted if the previous boolean is true.
+*
+* err = 3 Reciprocal condition number: Estimated componentwise
+* reciprocal condition number. Compared with the threshold
+* sqrt(n) * dlamch('Epsilon') to determine if the error
+* estimate is "guaranteed". These reciprocal condition
+* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
+* appropriately scaled matrix Z.
+* Let Z = S*(A*diag(x)), where x is the solution for the
+* current right-hand side and S scales each row of
+* A*diag(x) by a power of the radix so all absolute row
+* sums of Z are approximately 1.
+*
+* See Lapack Working Note 165 for further details and extra
+* cautions.
+*
+* NPARAMS (input) INTEGER
+* Specifies the number of parameters set in PARAMS. If .LE. 0, the
+* PARAMS array is never referenced and default values are used.
+*
+* PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
+* Specifies algorithm parameters. If an entry is .LT. 0.0, then
+* that entry will be filled with default value used for that
+* parameter. Only positions up to NPARAMS are accessed; defaults
+* are used for higher-numbered parameters.
+*
+* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
+* refinement or not.
+* Default: 1.0D+0
+* = 0.0 : No refinement is performed, and no error bounds are
+* computed.
+* = 1.0 : Use the extra-precise refinement algorithm.
+* (other values are reserved for future use)
+*
+* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
+* computations allowed for refinement.
+* Default: 10
+* Aggressive: Set to 100 to permit convergence using approximate
+* factorizations or factorizations other than LU. If
+* the factorization uses a technique other than
+* Gaussian elimination, the guarantees in
+* err_bnds_norm and err_bnds_comp may no longer be
+* trustworthy.
+*
+* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
+* will attempt to find a solution with small componentwise
+* relative error in the double-precision algorithm. Positive
+* is true, 0.0 is false.
+* Default: 1.0 (attempt componentwise convergence)
+*
+* WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
+*
+* IWORK (workspace) INTEGER array, dimension (N)
+*
+* INFO (output) INTEGER
+* = 0: Successful exit. The solution to every right-hand side is
+* guaranteed.
+* < 0: If INFO = -i, the i-th argument had an illegal value
+* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
+* has been completed, but the factor U is exactly singular, so
+* the solution and error bounds could not be computed. RCOND = 0
+* is returned.
+* = N+J: The solution corresponding to the Jth right-hand side is
+* not guaranteed. The solutions corresponding to other right-
+* hand sides K with K > J may not be guaranteed as well, but
+* only the first such right-hand side is reported. If a small
+* componentwise error is not requested (PARAMS(3) = 0.0) then
+* the Jth right-hand side is the first with a normwise error
+* bound that is not guaranteed (the smallest J such
+* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
+* the Jth right-hand side is the first with either a normwise or
+* componentwise error bound that is not guaranteed (the smallest
+* J such that either ERR_BNDS_NORM(J,1) = 0.0 or
+* ERR_BNDS_COMP(J,1) = 0.0). See the definition of
+* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
+* about all of the right-hand sides check ERR_BNDS_NORM or
+* ERR_BNDS_COMP.
+*
+* ==================================================================
+*
+* .. Parameters ..
+ DOUBLE PRECISION ZERO, ONE
+ PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
+ INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
+ INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
+ INTEGER CMP_ERR_I, PIV_GROWTH_I
+ PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
+ $ BERR_I = 3 )
+ PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
+ PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
+ $ PIV_GROWTH_I = 9 )
+* ..
+* .. Local Scalars ..
+ LOGICAL EQUIL, NOFACT, RCEQU
+ INTEGER INFEQU, J
+ DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
+* ..
+* .. External Functions ..
+ EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW
+ LOGICAL LSAME
+ DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
+* ..
+* .. External Subroutines ..
+ EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY,
+ $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC MAX, MIN
+* ..
+* .. Executable Statements ..
+*
+ INFO = 0
+ NOFACT = LSAME( FACT, 'N' )
+ EQUIL = LSAME( FACT, 'E' )
+ SMLNUM = DLAMCH( 'Safe minimum' )
+ BIGNUM = ONE / SMLNUM
+ IF( NOFACT .OR. EQUIL ) THEN
+ EQUED = 'N'
+ RCEQU = .FALSE.
+ ELSE
+ RCEQU = LSAME( EQUED, 'Y' )
+ ENDIF
+*
+* Default is failure. If an input parameter is wrong or
+* factorization fails, make everything look horrible. Only the
+* pivot growth is set here, the rest is initialized in ZPORFSX.
+*
+ RPVGRW = ZERO
+*
+* Test the input parameters. PARAMS is not tested until ZPORFSX.
+*
+ IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
+ $ LSAME( FACT, 'F' ) ) THEN
+ INFO = -1
+ ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
+ $ .NOT.LSAME( UPLO, 'L' ) ) THEN
+ INFO = -2
+ ELSE IF( N.LT.0 ) THEN
+ INFO = -3
+ ELSE IF( NRHS.LT.0 ) THEN
+ INFO = -4
+ ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
+ INFO = -6
+ ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
+ INFO = -8
+ ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
+ $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
+ INFO = -9
+ ELSE
+ IF ( RCEQU ) THEN
+ SMIN = BIGNUM
+ SMAX = ZERO
+ DO 10 J = 1, N
+ SMIN = MIN( SMIN, S( J ) )
+ SMAX = MAX( SMAX, S( J ) )
+ 10 CONTINUE
+ IF( SMIN.LE.ZERO ) THEN
+ INFO = -10
+ ELSE IF( N.GT.0 ) THEN
+ SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
+ ELSE
+ SCOND = ONE
+ END IF
+ END IF
+ IF( INFO.EQ.0 ) THEN
+ IF( LDB.LT.MAX( 1, N ) ) THEN
+ INFO = -12
+ ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
+ INFO = -14
+ END IF
+ END IF
+ END IF
+*
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'ZPOSVXX', -INFO )
+ RETURN
+ END IF
+*
+ IF( EQUIL ) THEN
+*
+* Compute row and column scalings to equilibrate the matrix A.
+*
+ CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
+ IF( INFEQU.EQ.0 ) THEN
+*
+* Equilibrate the matrix.
+*
+ CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
+ RCEQU = LSAME( EQUED, 'Y' )
+ END IF
+ END IF
+*
+* Scale the right-hand side.
+*
+ IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
+*
+ IF( NOFACT .OR. EQUIL ) THEN
+*
+* Compute the LU factorization of A.
+*
+ CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
+ CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
+*
+* Return if INFO is non-zero.
+*
+ IF( INFO.GT.0 ) THEN
+*
+* Pivot in column INFO is exactly 0
+* Compute the reciprocal pivot growth factor of the
+* leading rank-deficient INFO columns of A.
+*
+ RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK )
+ RETURN
+ END IF
+ END IF
+*
+* Compute the reciprocal pivot growth factor RPVGRW.
+*
+ RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK )
+*
+* Compute the solution matrix X.
+*
+ CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
+ CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
+*
+* Use iterative refinement to improve the computed solution and
+* compute error bounds and backward error estimates for it.
+*
+ CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
+ $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
+ $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
+
+*
+* Scale solutions.
+*
+ IF ( RCEQU ) THEN
+ CALL ZLASCL2( N, NRHS, S, X, LDX )
+ END IF
+*
+ RETURN
+*
+* End of ZPOSVXX
+*
+ END