ZPPSVX(3F) ZPPSVX(3F)
ZPPSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations A * X = B,
SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N Hermitian positive definite matrix
stored in packed format and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also provided.
The following steps are performed:
1. If FACT = 'E', real 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'* U , if UPLO = 'U', or
A = L * L', if UPLO = 'L',
where U is an upper triangular matrix, L is a lower triangular
matrix, and ' indicates conjugate transpose.
3. 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, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
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ZPPSVX(3F) ZPPSVX(3F)
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.
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, AFP
contains the factored form of A. If EQUED = 'Y', the matrix A
has been equilibrated with scaling factors given by S. AP and
AFP will not be modified. = 'N': The matrix A will be copied to
AFP and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AFP 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.
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix A,
packed columnwise in a linear array, except if FACT = 'F' and
EQUED = 'Y', then A must contain the equilibrated matrix
diag(S)*A*diag(S). The j-th column of A is stored in the array
AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for
1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
j<=i<=n. See below for further details. 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).
AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization A = U**H*U or A = L*L**H, in the same storage
format as A. If EQUED .ne. 'N', then AFP is the factored form of
the equilibrated matrix A.
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ZPPSVX(3F) ZPPSVX(3F)
If FACT = 'N', then AFP is an output argument and on exit returns
the triangular factor U or L from the Cholesky factorization A =
U**H*U or A = L*L**H of the original matrix A.
If FACT = 'E', then AFP is an output argument and on exit returns
the triangular factor U or L from the Cholesky factorization A =
U**H*U or A = L*L**H of the equilibrated matrix A (see the
description of AP for the form of the equilibrated matrix).
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'Y': Equilibration was done, 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 scale factors for A; not accessed if EQUED = 'N'. 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.
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 if EQUED = 'Y', A and B are
modified on exit, 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
The estimate of the reciprocal condition number of the matrix A
after equilibration (if done). 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, and the solution and error bounds are not
computed.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated 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)
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ZPPSVX(3F) ZPPSVX(3F)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) 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).
WORK (workspace) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) 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 and
error bounds could not be computed. = N+1: RCOND is less than
machine precision. The factorization has been completed, but the
matrix is singular to working precision, and the solution and
error bounds have not been computed.
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when N
= 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
ZPPSVX(3F) ZPPSVX(3F)
ZPPSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations A * X = B,
SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N Hermitian positive definite matrix
stored in packed format and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also provided.
The following steps are performed:
1. If FACT = 'E', real 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'* U , if UPLO = 'U', or
A = L * L', if UPLO = 'L',
where U is an upper triangular matrix, L is a lower triangular
matrix, and ' indicates conjugate transpose.
3. 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, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
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ZPPSVX(3F) ZPPSVX(3F)
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.
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, AFP
contains the factored form of A. If EQUED = 'Y', the matrix A
has been equilibrated with scaling factors given by S. AP and
AFP will not be modified. = 'N': The matrix A will be copied to
AFP and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AFP 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.
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix A,
packed columnwise in a linear array, except if FACT = 'F' and
EQUED = 'Y', then A must contain the equilibrated matrix
diag(S)*A*diag(S). The j-th column of A is stored in the array
AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for
1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
j<=i<=n. See below for further details. 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).
AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization A = U**H*U or A = L*L**H, in the same storage
format as A. If EQUED .ne. 'N', then AFP is the factored form of
the equilibrated matrix A.
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ZPPSVX(3F) ZPPSVX(3F)
If FACT = 'N', then AFP is an output argument and on exit returns
the triangular factor U or L from the Cholesky factorization A =
U**H*U or A = L*L**H of the original matrix A.
If FACT = 'E', then AFP is an output argument and on exit returns
the triangular factor U or L from the Cholesky factorization A =
U**H*U or A = L*L**H of the equilibrated matrix A (see the
description of AP for the form of the equilibrated matrix).
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'Y': Equilibration was done, 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 scale factors for A; not accessed if EQUED = 'N'. 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.
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 if EQUED = 'Y', A and B are
modified on exit, 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
The estimate of the reciprocal condition number of the matrix A
after equilibration (if done). 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, and the solution and error bounds are not
computed.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated 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)
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ZPPSVX(3F) ZPPSVX(3F)
divided by the magnitude of the largest element in X(j). The
estimate is as reliable as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) 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).
WORK (workspace) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) 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 and
error bounds could not be computed. = N+1: RCOND is less than
machine precision. The factorization has been completed, but the
matrix is singular to working precision, and the solution and
error bounds have not been computed.
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when N
= 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
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