Man page - ppsvx(3)

Packages contains this manual

Package: liblapack-doc
apt-get install liblapack-doc

Manuals in package:

Documentations in package:

Manual

ppsvx

NAME

ppsvx - ppsvx: factor and solve, expert

SYNOPSIS

Functions

subroutine cppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

subroutine cppsvx (character fact, character uplo, integer n, integer nrhs,complex, dimension( * ) ap, complex, dimension( * ) afp, characterequed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integerldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real,dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * )work, real, dimension( * ) rwork, integer info)

CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

CPPSVX 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.

Description:

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**H * U , if UPLO = ’U’, or
A = L * L**H, if UPLO = ’L’,
where U is an upper triangular matrix, L is a lower triangular
matrix, and **H indicates conjugate transpose.

3. If the leading principal minor of order i is not positive,
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.

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
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.

Parameters

FACT

FACT is 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

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

AP is COMPLEX 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

AFP is COMPLEX 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.

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

EQUED is 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 is REAL 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 is COMPLEX 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

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, 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

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND

RCOND is REAL
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.

FERR

FERR is REAL 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) 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

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).

WORK

WORK is COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL array, dimension (N)

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 principal minor of order i of A
is not positive, 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.

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 ]

subroutine dppsvx (character fact, character uplo, integer n, integer nrhs,double precision, dimension( * ) ap, double precision, dimension( * )afp, character equed, double precision, dimension( * ) s, doubleprecision, dimension( ldb, * ) b, integer ldb, double precision,dimension( ldx, * ) x, integer ldx, double precision rcond, doubleprecision, dimension( * ) ferr, double precision, dimension( * ) berr,double precision, dimension( * ) work, integer, dimension( * ) iwork,integer info)

DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

DPPSVX uses the Cholesky factorization A = U**T*U or A = L*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 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.

Description:

The following steps are performed:

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.

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
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.

Parameters

FACT

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric 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

AFP is DOUBLE PRECISION 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**T*U or A = L*L**T, in the same storage
format as A. If EQUED .ne. ’N’, then AFP is the factored
form of the equilibrated matrix A.

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**T * U or A = L * L**T 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**T * U or A = L * L**T of the equilibrated
matrix A (see the description of AP for the form of the
equilibrated matrix).

EQUED

S is 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 is DOUBLE PRECISION 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

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, 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

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND

RCOND is 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.

FERR

FERR is 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) 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

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).

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The packed storage scheme is illustrated by the following example
when N = 4, UPLO = ’U’:

Two-dimensional storage of the symmetric 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 ]

subroutine sppsvx (character fact, character uplo, integer n, integer nrhs,real, dimension( * ) ap, real, dimension( * ) afp, character equed,real, dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real,dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * )ferr, real, dimension( * ) berr, real, dimension( * ) work, integer,dimension( * ) iwork, integer info)

SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

SPPSVX uses the Cholesky factorization A = U**T*U or A = L*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 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.

Description:

The following steps are performed:

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
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.

Parameters

FACT

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric 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

AFP is REAL 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**T*U or A = L*L**T, in the same storage
format as A. If EQUED .ne. ’N’, then AFP is the factored
form of the equilibrated matrix A.

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**T * U or A = L * L**T of the original
matrix A.

EQUED

B is REAL 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

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is REAL array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, 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

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND

FERR

BERR

WORK

WORK is REAL array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The packed storage scheme is illustrated by the following example
when N = 4, UPLO = ’U’:

Two-dimensional storage of the symmetric 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 ]

subroutine zppsvx (character fact, character uplo, integer n, integer nrhs,complex16, dimension( ) ap, complex16, dimension( ) afp,character equed, double precision, dimension( * ) s, complex16,dimension( ldb, ) b, integer ldb, complex16, dimension( ldx, ) x,integer ldx, double precision rcond, double precision, dimension( * )ferr, double precision, dimension( * ) berr, complex16, dimension( )work, double precision, dimension( * ) rwork, integer info)

ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

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.

Description:

The following steps are performed:

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
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.

Parameters

FACT

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

AP is 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

AFP is 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.

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

B is 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

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, 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

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND

FERR

BERR

WORK

WORK is COMPLEX*16 array, dimension (2*N)

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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 ]

Author

Generated automatically by Doxygen for LAPACK from the source code.