382 lines
14 KiB
FortranFixed
382 lines
14 KiB
FortranFixed
SUBROUTINE CPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
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$ X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER EQUED, FACT, UPLO
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INTEGER INFO, LDB, LDX, N, NRHS
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REAL RCOND
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* ..
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* .. Array Arguments ..
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REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
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COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
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$ X( LDX, * )
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* ..
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*
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* Purpose
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* =======
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*
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* CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
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* compute the solution to a complex system of linear equations
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* A * X = B,
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* where A is an N-by-N Hermitian positive definite matrix stored in
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* packed format and X and B are N-by-NRHS matrices.
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*
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* Error bounds on the solution and a condition estimate are also
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* provided.
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*
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* Description
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* ===========
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*
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* The following steps are performed:
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*
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* 1. If FACT = 'E', real scaling factors are computed to equilibrate
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* the system:
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* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
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* Whether or not the system will be equilibrated depends on the
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* scaling of the matrix A, but if equilibration is used, A is
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* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
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*
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* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
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* factor the matrix A (after equilibration if FACT = 'E') as
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* A = U'* U , if UPLO = 'U', or
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* A = L * L', if UPLO = 'L',
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* where U is an upper triangular matrix, L is a lower triangular
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* matrix, and ' indicates conjugate transpose.
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*
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* 3. If the leading i-by-i principal minor is not positive definite,
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* then the routine returns with INFO = i. Otherwise, the factored
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* form of A is used to estimate the condition number of the matrix
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* A. If the reciprocal of the condition number is less than machine
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* precision, INFO = N+1 is returned as a warning, but the routine
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* still goes on to solve for X and compute error bounds as
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* described below.
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*
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* 4. The system of equations is solved for X using the factored form
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* of A.
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*
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* 5. Iterative refinement is applied to improve the computed solution
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* matrix and calculate error bounds and backward error estimates
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* for it.
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*
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* 6. If equilibration was used, the matrix X is premultiplied by
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* diag(S) so that it solves the original system before
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* equilibration.
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*
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* Arguments
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* =========
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*
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* FACT (input) CHARACTER*1
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* Specifies whether or not the factored form of the matrix A is
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* supplied on entry, and if not, whether the matrix A should be
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* equilibrated before it is factored.
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* = 'F': On entry, AFP contains the factored form of A.
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* If EQUED = 'Y', the matrix A has been equilibrated
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* with scaling factors given by S. AP and AFP will not
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* be modified.
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* = 'N': The matrix A will be copied to AFP and factored.
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* = 'E': The matrix A will be equilibrated if necessary, then
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* copied to AFP and factored.
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*
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* UPLO (input) CHARACTER*1
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* = 'U': Upper triangle of A is stored;
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* = 'L': Lower triangle of A is stored.
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*
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* N (input) INTEGER
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* The number of linear equations, i.e., the order of the
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* matrix A. N >= 0.
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*
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* NRHS (input) INTEGER
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* The number of right hand sides, i.e., the number of columns
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* of the matrices B and X. NRHS >= 0.
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*
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* AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
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* On entry, the upper or lower triangle of the Hermitian matrix
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* A, packed columnwise in a linear array, except if FACT = 'F'
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* and EQUED = 'Y', then A must contain the equilibrated matrix
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* diag(S)*A*diag(S). The j-th column of A is stored in the
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* array AP as follows:
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* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
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* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
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* See below for further details. A is not modified if
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* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
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*
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* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
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* diag(S)*A*diag(S).
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*
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* AFP (input or output) COMPLEX array, dimension (N*(N+1)/2)
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* If FACT = 'F', then AFP is an input argument and on entry
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* contains the triangular factor U or L from the Cholesky
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* factorization A = U**H*U or A = L*L**H, in the same storage
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* format as A. If EQUED .ne. 'N', then AFP is the factored
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* form of the equilibrated matrix A.
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*
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* If FACT = 'N', then AFP is an output argument and on exit
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* returns the triangular factor U or L from the Cholesky
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* factorization A = U**H*U or A = L*L**H of the original
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* matrix A.
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*
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* If FACT = 'E', then AFP is an output argument and on exit
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* returns the triangular factor U or L from the Cholesky
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* factorization A = U**H*U or A = L*L**H of the equilibrated
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* matrix A (see the description of AP for the form of the
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* equilibrated matrix).
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*
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* EQUED (input or output) CHARACTER*1
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* Specifies the form of equilibration that was done.
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* = 'N': No equilibration (always true if FACT = 'N').
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* = 'Y': Equilibration was done, i.e., A has been replaced by
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* diag(S) * A * diag(S).
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* EQUED is an input argument if FACT = 'F'; otherwise, it is an
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* output argument.
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*
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* S (input or output) REAL array, dimension (N)
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* The scale factors for A; not accessed if EQUED = 'N'. S is
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* an input argument if FACT = 'F'; otherwise, S is an output
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* argument. If FACT = 'F' and EQUED = 'Y', each element of S
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* must be positive.
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*
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* B (input/output) COMPLEX array, dimension (LDB,NRHS)
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* On entry, the N-by-NRHS right hand side matrix B.
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* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
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* B is overwritten by diag(S) * B.
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,N).
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*
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* X (output) COMPLEX array, dimension (LDX,NRHS)
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* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
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* the original system of equations. Note that if EQUED = 'Y',
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* A and B are modified on exit, and the solution to the
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* equilibrated system is inv(diag(S))*X.
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*
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* LDX (input) INTEGER
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* The leading dimension of the array X. LDX >= max(1,N).
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*
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* RCOND (output) REAL
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* The estimate of the reciprocal condition number of the matrix
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* A after equilibration (if done). If RCOND is less than the
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* machine precision (in particular, if RCOND = 0), the matrix
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* is singular to working precision. This condition is
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* indicated by a return code of INFO > 0.
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*
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* FERR (output) REAL array, dimension (NRHS)
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* The estimated forward error bound for each solution vector
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* X(j) (the j-th column of the solution matrix X).
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* If XTRUE is the true solution corresponding to X(j), FERR(j)
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* is an estimated upper bound for the magnitude of the largest
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* element in (X(j) - XTRUE) divided by the magnitude of the
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* largest element in X(j). The estimate is as reliable as
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* the estimate for RCOND, and is almost always a slight
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* overestimate of the true error.
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*
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* BERR (output) REAL array, dimension (NRHS)
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* The componentwise relative backward error of each solution
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* vector X(j) (i.e., the smallest relative change in
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* any element of A or B that makes X(j) an exact solution).
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*
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* WORK (workspace) COMPLEX array, dimension (2*N)
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*
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* RWORK (workspace) REAL array, dimension (N)
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value
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* > 0: if INFO = i, and i is
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* <= N: the leading minor of order i of A is
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* not positive definite, so the factorization
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* could not be completed, and the solution has not
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* been computed. RCOND = 0 is returned.
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* = N+1: U is nonsingular, but RCOND is less than machine
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* precision, meaning that the matrix is singular
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* to working precision. Nevertheless, the
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* solution and error bounds are computed because
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* there are a number of situations where the
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* computed solution can be more accurate than the
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* value of RCOND would suggest.
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*
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* Further Details
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* ===============
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*
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* The packed storage scheme is illustrated by the following example
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* when N = 4, UPLO = 'U':
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*
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* Two-dimensional storage of the Hermitian matrix A:
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*
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* a11 a12 a13 a14
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* a22 a23 a24
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* a33 a34 (aij = conjg(aji))
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* a44
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*
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* Packed storage of the upper triangle of A:
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*
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* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL EQUIL, NOFACT, RCEQU
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INTEGER I, INFEQU, J
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REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL CLANHP, SLAMCH
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EXTERNAL LSAME, CLANHP, SLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL CCOPY, CLACPY, CLAQHP, CPPCON, CPPEQU, CPPRFS,
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$ CPPTRF, CPPTRS, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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NOFACT = LSAME( FACT, 'N' )
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EQUIL = LSAME( FACT, 'E' )
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IF( NOFACT .OR. EQUIL ) THEN
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EQUED = 'N'
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RCEQU = .FALSE.
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ELSE
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RCEQU = LSAME( EQUED, 'Y' )
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SMLNUM = SLAMCH( 'Safe minimum' )
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BIGNUM = ONE / SMLNUM
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END IF
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*
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* Test the input parameters.
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*
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IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
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$ THEN
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INFO = -1
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ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
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$ THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -4
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ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
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$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
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INFO = -7
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ELSE
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IF( RCEQU ) THEN
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SMIN = BIGNUM
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SMAX = ZERO
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DO 10 J = 1, N
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SMIN = MIN( SMIN, S( J ) )
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SMAX = MAX( SMAX, S( J ) )
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10 CONTINUE
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IF( SMIN.LE.ZERO ) THEN
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INFO = -8
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ELSE IF( N.GT.0 ) THEN
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SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
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ELSE
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SCOND = ONE
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -10
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -12
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END IF
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CPPSVX', -INFO )
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RETURN
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END IF
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*
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IF( EQUIL ) THEN
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*
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* Compute row and column scalings to equilibrate the matrix A.
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*
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CALL CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
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IF( INFEQU.EQ.0 ) THEN
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*
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* Equilibrate the matrix.
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*
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CALL CLAQHP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
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RCEQU = LSAME( EQUED, 'Y' )
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END IF
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END IF
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*
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* Scale the right-hand side.
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*
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IF( RCEQU ) THEN
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DO 30 J = 1, NRHS
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DO 20 I = 1, N
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B( I, J ) = S( I )*B( I, J )
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20 CONTINUE
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30 CONTINUE
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END IF
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*
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IF( NOFACT .OR. EQUIL ) THEN
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*
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* Compute the Cholesky factorization A = U'*U or A = L*L'.
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*
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CALL CCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
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CALL CPPTRF( UPLO, N, AFP, INFO )
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*
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* Return if INFO is non-zero.
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*
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IF( INFO.GT.0 )THEN
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RCOND = ZERO
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RETURN
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END IF
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END IF
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*
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* Compute the norm of the matrix A.
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*
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ANORM = CLANHP( 'I', UPLO, N, AP, RWORK )
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*
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* Compute the reciprocal of the condition number of A.
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*
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CALL CPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, RWORK, INFO )
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*
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* Compute the solution matrix X.
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*
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CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
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CALL CPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
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*
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* Use iterative refinement to improve the computed solution and
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* compute error bounds and backward error estimates for it.
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*
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CALL CPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
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$ WORK, RWORK, INFO )
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*
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* Transform the solution matrix X to a solution of the original
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* system.
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*
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IF( RCEQU ) THEN
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DO 50 J = 1, NRHS
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DO 40 I = 1, N
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X( I, J ) = S( I )*X( I, J )
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40 CONTINUE
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50 CONTINUE
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DO 60 J = 1, NRHS
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FERR( J ) = FERR( J ) / SCOND
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60 CONTINUE
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END IF
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*
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* Set INFO = N+1 if the matrix is singular to working precision.
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*
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IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
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$ INFO = N + 1
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*
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RETURN
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*
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* End of CPPSVX
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*
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END
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