.TH "pprfs" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
pprfs \- pprfs: iterative refinement
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcpprfs\fP (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, rwork, info)"
.br
.RI "\fBCPPRFS\fP "
.ti -1c
.RI "subroutine \fBdpprfs\fP (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info)"
.br
.RI "\fBDPPRFS\fP "
.ti -1c
.RI "subroutine \fBspprfs\fP (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info)"
.br
.RI "\fBSPPRFS\fP "
.ti -1c
.RI "subroutine \fBzpprfs\fP (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, rwork, info)"
.br
.RI "\fBZPPRFS\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cpprfs (character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)"

.PP
\fBCPPRFS\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> CPPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian positive definite
!> and packed, and provides error bounds and backward error estimates
!> for the solution\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
!>          NRHS is INTEGER
!>          The number of right hand sides, i\&.e\&., the number of columns
!>          of the matrices B and X\&.  NRHS >= 0\&.
!> 
.fi
.PP
.br
\fIAP\fP 
.PP
.nf
!>          AP is COMPLEX array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the Hermitian matrix A, packed
!>          columnwise in a linear array\&.  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\&.
!> 
.fi
.PP
.br
\fIAFP\fP 
.PP
.nf
!>          AFP is COMPLEX array, dimension (N*(N+1)/2)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**H*U or A = L*L**H, as computed by SPPTRF/CPPTRF,
!>          packed columnwise in a linear array in the same format as A
!>          (see AP)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          B is COMPLEX array, dimension (LDB,NRHS)
!>          The right hand side matrix B\&.
!> 
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
!>          LDB is INTEGER
!>          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIX\fP 
.PP
.nf
!>          X is COMPLEX array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by CPPTRS\&.
!>          On exit, the improved solution matrix X\&.
!> 
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
!>          LDX is INTEGER
!>          The leading dimension of the array X\&.  LDX >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIFERR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIBERR\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX array, dimension (2*N)
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is REAL array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  ITMAX is the maximum number of steps of iterative refinement\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB169\fP of file \fBcpprfs\&.f\fP\&.
.SS "subroutine dpprfs (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBDPPRFS\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> DPPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric positive definite
!> and packed, and provides error bounds and backward error estimates
!> for the solution\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
!>          NRHS is INTEGER
!>          The number of right hand sides, i\&.e\&., the number of columns
!>          of the matrices B and X\&.  NRHS >= 0\&.
!> 
.fi
.PP
.br
\fIAP\fP 
.PP
.nf
!>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array\&.  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\&.
!> 
.fi
.PP
.br
\fIAFP\fP 
.PP
.nf
!>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
!>          packed columnwise in a linear array in the same format as A
!>          (see AP)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
!>          The right hand side matrix B\&.
!> 
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
!>          LDB is INTEGER
!>          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIX\fP 
.PP
.nf
!>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by DPPTRS\&.
!>          On exit, the improved solution matrix X\&.
!> 
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
!>          LDX is INTEGER
!>          The leading dimension of the array X\&.  LDX >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIFERR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIBERR\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (3*N)
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  ITMAX is the maximum number of steps of iterative refinement\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB169\fP of file \fBdpprfs\&.f\fP\&.
.SS "subroutine spprfs (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)"

.PP
\fBSPPRFS\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> SPPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric positive definite
!> and packed, and provides error bounds and backward error estimates
!> for the solution\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
!>          NRHS is INTEGER
!>          The number of right hand sides, i\&.e\&., the number of columns
!>          of the matrices B and X\&.  NRHS >= 0\&.
!> 
.fi
.PP
.br
\fIAP\fP 
.PP
.nf
!>          AP is REAL array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the symmetric matrix A, packed
!>          columnwise in a linear array\&.  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\&.
!> 
.fi
.PP
.br
\fIAFP\fP 
.PP
.nf
!>          AFP is REAL array, dimension (N*(N+1)/2)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**T*U or A = L*L**T, as computed by SPPTRF/CPPTRF,
!>          packed columnwise in a linear array in the same format as A
!>          (see AP)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          B is REAL array, dimension (LDB,NRHS)
!>          The right hand side matrix B\&.
!> 
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
!>          LDB is INTEGER
!>          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIX\fP 
.PP
.nf
!>          X is REAL array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by SPPTRS\&.
!>          On exit, the improved solution matrix X\&.
!> 
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
!>          LDX is INTEGER
!>          The leading dimension of the array X\&.  LDX >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIFERR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIBERR\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (3*N)
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  ITMAX is the maximum number of steps of iterative refinement\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB169\fP of file \fBspprfs\&.f\fP\&.
.SS "subroutine zpprfs (character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)"

.PP
\fBZPPRFS\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZPPRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is Hermitian positive definite
!> and packed, and provides error bounds and backward error estimates
!> for the solution\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
!>          NRHS is INTEGER
!>          The number of right hand sides, i\&.e\&., the number of columns
!>          of the matrices B and X\&.  NRHS >= 0\&.
!> 
.fi
.PP
.br
\fIAP\fP 
.PP
.nf
!>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The upper or lower triangle of the Hermitian matrix A, packed
!>          columnwise in a linear array\&.  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\&.
!> 
.fi
.PP
.br
\fIAFP\fP 
.PP
.nf
!>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          The triangular factor U or L from the Cholesky factorization
!>          A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF,
!>          packed columnwise in a linear array in the same format as A
!>          (see AP)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          B is COMPLEX*16 array, dimension (LDB,NRHS)
!>          The right hand side matrix B\&.
!> 
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
!>          LDB is INTEGER
!>          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIX\fP 
.PP
.nf
!>          X is COMPLEX*16 array, dimension (LDX,NRHS)
!>          On entry, the solution matrix X, as computed by ZPPTRS\&.
!>          On exit, the improved solution matrix X\&.
!> 
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
!>          LDX is INTEGER
!>          The leading dimension of the array X\&.  LDX >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIFERR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIBERR\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension (2*N)
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is DOUBLE PRECISION array, dimension (N)
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
!>  ITMAX is the maximum number of steps of iterative refinement\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB169\fP of file \fBzpprfs\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.