.TH "posv_mixed" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
posv_mixed \- posv: factor and solve, mixed precision
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBdsposv\fP (uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, iter, info)"
.br
.RI "\fB DSPOSV computes the solution to system of linear equations A * X = B for PO matrices\fP "
.ti -1c
.RI "subroutine \fBzcposv\fP (uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, rwork, iter, info)"
.br
.RI "\fB ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine dsposv (character uplo, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( n, * ) work, real, dimension( * ) swork, integer iter, integer info)"

.PP
\fB DSPOSV computes the solution to system of linear equations A * X = B for PO matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DSPOSV computes the solution to a real 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\&.

 DSPOSV first attempts to factorize the matrix in SINGLE PRECISION
 and use this factorization within an iterative refinement procedure
 to produce a solution with DOUBLE PRECISION normwise backward error
 quality (see below)\&. If the approach fails the method switches to a
 DOUBLE PRECISION factorization and solve\&.

 The iterative refinement is not going to be a winning strategy if
 the ratio SINGLE PRECISION performance over DOUBLE PRECISION
 performance is too small\&. A reasonable strategy should take the
 number of right-hand sides and the size of the matrix into account\&.
 This might be done with a call to ILAENV in the future\&. Up to now, we
 always try iterative refinement\&.

 The iterative refinement process is stopped if
     ITER > ITERMAX
 or for all the RHS we have:
     RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
 where
     o ITER is the number of the current iteration in the iterative
       refinement process
     o RNRM is the infinity-norm of the residual
     o XNRM is the infinity-norm of the solution
     o ANRM is the infinity-operator-norm of the matrix A
     o EPS is the machine epsilon returned by DLAMCH('Epsilon')
 The value ITERMAX and BWDMAX are fixed to 30 and 1\&.0D+00
 respectively\&.
.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 number of linear equations, i\&.e\&., 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 matrix B\&.  NRHS >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array,
          dimension (LDA,N)
          On entry, the symmetric matrix A\&.  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\&.
          On exit, if iterative refinement has been successfully used
          (INFO = 0 and ITER >= 0, see description below), then A is
          unchanged, if double precision factorization has been used
          (INFO = 0 and ITER < 0, see description below), then the
          array A contains the factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS 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)
          If INFO = 0, the N-by-NRHS 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
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (N,NRHS)
          This array is used to hold the residual vectors\&.
.fi
.PP
.br
\fISWORK\fP 
.PP
.nf
          SWORK is REAL array, dimension (N*(N+NRHS))
          This array is used to use the single precision matrix and the
          right-hand sides or solutions in single precision\&.
.fi
.PP
.br
\fIITER\fP 
.PP
.nf
          ITER is INTEGER
          < 0: iterative refinement has failed, double precision
               factorization has been performed
               -1 : the routine fell back to full precision for
                    implementation- or machine-specific reasons
               -2 : narrowing the precision induced an overflow,
                    the routine fell back to full precision
               -3 : failure of SPOTRF
               -31: stop the iterative refinement after the 30th
                    iterations
          > 0: iterative refinement has been successfully used\&.
               Returns the number of iterations
.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
          > 0:  if INFO = i, the leading principal minor of order i
                of (DOUBLE PRECISION) A is not positive, so the
                factorization could not be completed, and the solution
                has not been computed\&.
.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

.SS "subroutine zcposv (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, complex*16, dimension( n, * ) work, complex, dimension( * ) swork, double precision, dimension( * ) rwork, integer iter, integer info)"

.PP
\fB ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZCPOSV computes the solution to a complex system of linear equations
    A * X = B,
 where A is an N-by-N Hermitian positive definite matrix and X and B
 are N-by-NRHS matrices\&.

 ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
 factorization within an iterative refinement procedure to produce a
 solution with COMPLEX*16 normwise backward error quality (see below)\&.
 If the approach fails the method switches to a COMPLEX*16
 factorization and solve\&.

 The iterative refinement is not going to be a winning strategy if
 the ratio COMPLEX performance over COMPLEX*16 performance is too
 small\&. A reasonable strategy should take the number of right-hand
 sides and the size of the matrix into account\&. This might be done
 with a call to ILAENV in the future\&. Up to now, we always try
 iterative refinement\&.

 The iterative refinement process is stopped if
     ITER > ITERMAX
 or for all the RHS we have:
     RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
 where
     o ITER is the number of the current iteration in the iterative
       refinement process
     o RNRM is the infinity-norm of the residual
     o XNRM is the infinity-norm of the solution
     o ANRM is the infinity-operator-norm of the matrix A
     o EPS is the machine epsilon returned by DLAMCH('Epsilon')
 The value ITERMAX and BWDMAX are fixed to 30 and 1\&.0D+00
 respectively\&.
.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 number of linear equations, i\&.e\&., 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 matrix B\&.  NRHS >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array,
          dimension (LDA,N)
          On entry, the Hermitian matrix A\&. 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\&.

          Note that the imaginary parts of the diagonal
          elements need not be set and are assumed to be zero\&.

          On exit, if iterative refinement has been successfully used
          (INFO = 0 and ITER >= 0, see description below), then A is
          unchanged, if double precision factorization has been used
          (INFO = 0 and ITER < 0, see description below), then the
          array A contains the factor U or L from the Cholesky
          factorization A = U**H*U or A = L*L**H\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX*16 array, dimension (LDB,NRHS)
          The N-by-NRHS 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)
          If INFO = 0, the N-by-NRHS 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
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (N,NRHS)
          This array is used to hold the residual vectors\&.
.fi
.PP
.br
\fISWORK\fP 
.PP
.nf
          SWORK is COMPLEX array, dimension (N*(N+NRHS))
          This array is used to use the single precision matrix and the
          right-hand sides or solutions in single precision\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIITER\fP 
.PP
.nf
          ITER is INTEGER
          < 0: iterative refinement has failed, COMPLEX*16
               factorization has been performed
               -1 : the routine fell back to full precision for
                    implementation- or machine-specific reasons
               -2 : narrowing the precision induced an overflow,
                    the routine fell back to full precision
               -3 : failure of CPOTRF
               -31: stop the iterative refinement after the 30th
                    iterations
          > 0: iterative refinement has been successfully used\&.
               Returns the number of iterations
.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
          > 0:  if INFO = i, the leading principal minor of order i
                of (COMPLEX*16) A is not positive, so the factorization
                could not be completed, and the solution has not been
                computed\&.
.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

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.