.TH "doublePOcomputational" 3 "Sun Nov 27 2022" "Version 3.11.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
doublePOcomputational \- double
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "double precision function \fBdla_porcond\fP (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK)"
.br
.RI "\fBDLA_PORCOND\fP estimates the Skeel condition number for a symmetric positive-definite matrix\&. "
.ti -1c
.RI "subroutine \fBdla_porfsx_extended\fP (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)"
.br
.RI "\fBDLA_PORFSX_EXTENDED\fP improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution\&. "
.ti -1c
.RI "double precision function \fBdla_porpvgrw\fP (UPLO, NCOLS, A, LDA, AF, LDAF, WORK)"
.br
.RI "\fBDLA_PORPVGRW\fP computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix\&. "
.ti -1c
.RI "subroutine \fBdpocon\fP (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)"
.br
.RI "\fBDPOCON\fP "
.ti -1c
.RI "subroutine \fBdpoequ\fP (N, A, LDA, S, SCOND, AMAX, INFO)"
.br
.RI "\fBDPOEQU\fP "
.ti -1c
.RI "subroutine \fBdpoequb\fP (N, A, LDA, S, SCOND, AMAX, INFO)"
.br
.RI "\fBDPOEQUB\fP "
.ti -1c
.RI "subroutine \fBdporfs\fP (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)"
.br
.RI "\fBDPORFS\fP "
.ti -1c
.RI "subroutine \fBdporfsx\fP (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)"
.br
.RI "\fBDPORFSX\fP "
.ti -1c
.RI "subroutine \fBdpotf2\fP (UPLO, N, A, LDA, INFO)"
.br
.RI "\fBDPOTF2\fP computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm)\&. "
.ti -1c
.RI "subroutine \fBdpotrf\fP (UPLO, N, A, LDA, INFO)"
.br
.RI "\fBDPOTRF\fP "
.ti -1c
.RI "recursive subroutine \fBdpotrf2\fP (UPLO, N, A, LDA, INFO)"
.br
.RI "\fBDPOTRF2\fP "
.ti -1c
.RI "subroutine \fBdpotri\fP (UPLO, N, A, LDA, INFO)"
.br
.RI "\fBDPOTRI\fP "
.ti -1c
.RI "subroutine \fBdpotrs\fP (UPLO, N, NRHS, A, LDA, B, LDB, INFO)"
.br
.RI "\fBDPOTRS\fP "
.in -1c
.SH "Detailed Description"
.PP
This is the group of double computational functions for PO matrices
.SH "Function Documentation"
.PP
.SS "double precision function dla_porcond (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK)"
.PP
\fBDLA_PORCOND\fP estimates the Skeel condition number for a symmetric positive-definite matrix\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DLA_PORCOND Estimates the Skeel condition number of op(A) * op2(C)
where op2 is determined by CMODE as follows
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
The Skeel condition number cond(A) = norminf( |inv(A)||A| )
is computed by computing scaling factors R such that
diag(R)*A*op2(C) is row equilibrated and computing the standard
infinity-norm condition number\&.
.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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAF\fP
.PP
.nf
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDAF\fP
.PP
.nf
LDAF is INTEGER
The leading dimension of the array AF\&. LDAF >= max(1,N)\&.
.fi
.PP
.br
\fICMODE\fP
.PP
.nf
CMODE is INTEGER
Determines op2(C) in the formula op(A) * op2(C) as follows:
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is DOUBLE PRECISION array, dimension (N)
The vector C in the formula op(A) * op2(C)\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: Successful exit\&.
i > 0: The ith argument is invalid\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (3*N)\&.
Workspace\&.
.fi
.PP
.br
\fIIWORK\fP
.PP
.nf
IWORK is INTEGER array, dimension (N)\&.
Workspace\&.
.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 dla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, double precision, dimension( * ) RES, double precision, dimension(*) AYB, double precision, dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)"
.PP
\fBDLA_PORFSX_EXTENDED\fP improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DLA_PORFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution\&.
This subroutine is called by DPORFSX to perform iterative refinement\&.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible\&. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds\&. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIPREC_TYPE\fP
.PP
.nf
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement\&.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
.fi
.PP
.br
\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\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAF\fP
.PP
.nf
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDAF\fP
.PP
.nf
LDAF is INTEGER
The leading dimension of the array AF\&. LDAF >= max(1,N)\&.
.fi
.PP
.br
\fICOLEQU\fP
.PP
.nf
COLEQU is LOGICAL
If \&.TRUE\&. then column equilibration was done to A before calling
this routine\&. This is needed to compute the solution and error
bounds correctly\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A\&. If COLEQU = \&.FALSE\&., C
is not accessed\&. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates\&.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows\&. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable\&.
.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
\fIY\fP
.PP
.nf
Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by DPOTRS\&.
On exit, the improved solution matrix Y\&.
.fi
.PP
.br
\fILDY\fP
.PP
.nf
LDY is INTEGER
The leading dimension of the array Y\&. LDY >= max(1,N)\&.
.fi
.PP
.br
\fIBERR_OUT\fP
.PP
.nf
BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z\&. This is computed by DLA_LIN_BERR\&.
.fi
.PP
.br
\fIN_NORMS\fP
.PP
.nf
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP)\&.
If N_NORMS >= 1 return normwise error bounds\&.
If N_NORMS >= 2 return componentwise error bounds\&.
.fi
.PP
.br
\fIERR_BNDS_NORM\fP
.PP
.nf
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below\&. There currently are up to three pieces of information
returned\&.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side\&.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean\&. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon')\&.
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon')\&. This error bound should only
be trusted if the previous boolean is true\&.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number\&. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'\&. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z\&.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1\&.
This subroutine is only responsible for setting the second field
above\&.
See Lapack Working Note 165 for further details and extra
cautions\&.
.fi
.PP
.br
\fIERR_BNDS_COMP\fP
.PP
.nf
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below\&. There currently are up to three
pieces of information returned for each right-hand side\&. If
componentwise accuracy is not requested (PARAMS(3) = 0\&.0), then
ERR_BNDS_COMP is not accessed\&. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned\&.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side\&.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean\&. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon')\&.
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon')\&. This error bound should only
be trusted if the previous boolean is true\&.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number\&. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is 'guaranteed'\&. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z\&.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1\&.
This subroutine is only responsible for setting the second field
above\&.
See Lapack Working Note 165 for further details and extra
cautions\&.
.fi
.PP
.br
\fIRES\fP
.PP
.nf
RES is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate residual\&.
.fi
.PP
.br
\fIAYB\fP
.PP
.nf
AYB is DOUBLE PRECISION array, dimension (N)
Workspace\&. This can be the same workspace passed for Y_TAIL\&.
.fi
.PP
.br
\fIDY\fP
.PP
.nf
DY is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate solution\&.
.fi
.PP
.br
\fIY_TAIL\fP
.PP
.nf
Y_TAIL is DOUBLE PRECISION array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution\&.
.fi
.PP
.br
\fIRCOND\fP
.PP
.nf
RCOND is DOUBLE PRECISION
Reciprocal scaled condition number\&. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done)\&. If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision\&. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned\&.
.fi
.PP
.br
\fIITHRESH\fP
.PP
.nf
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement\&. The default is 10\&. For 'aggressive' set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU\&. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy\&.
.fi
.PP
.br
\fIRTHRESH\fP
.PP
.nf
RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing\&. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z\&. RTHRESH satisfies 0 < RTHRESH <= 1\&. The
default value is 0\&.5\&. For 'aggressive' set to 0\&.9 to permit
convergence on extremely ill-conditioned matrices\&. See LAWN 165
for more details\&.
.fi
.PP
.br
\fIDZ_UB\fP
.PP
.nf
DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence\&.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB\&. The default value
is 0\&.25, requiring the first bit to be stable\&. See LAWN 165 for
more details\&.
.fi
.PP
.br
\fIIGNORE_CWISE\fP
.PP
.nf
IGNORE_CWISE is LOGICAL
If \&.TRUE\&. then ignore componentwise convergence\&. Default value
is \&.FALSE\&.\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: Successful exit\&.
< 0: if INFO = -i, the ith argument to DPOTRS had an illegal
value
.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 "double precision function dla_porpvgrw (character*1 UPLO, integer NCOLS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * ) WORK)"
.PP
\fBDLA_PORPVGRW\fP computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DLA_PORPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U)\&. The 'max absolute element' norm is used\&. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor\&. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable\&.
.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
\fINCOLS\fP
.PP
.nf
NCOLS is INTEGER
The number of columns of the matrix A\&. NCOLS >= 0\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAF\fP
.PP
.nf
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDAF\fP
.PP
.nf
LDAF is INTEGER
The leading dimension of the array AF\&. LDAF >= max(1,N)\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (2*N)
.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 dpocon (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)"
.PP
\fBDPOCON\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF\&.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A)))\&.
.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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIANORM\fP
.PP
.nf
ANORM is DOUBLE PRECISION
The 1-norm (or infinity-norm) of the symmetric matrix A\&.
.fi
.PP
.br
\fIRCOND\fP
.PP
.nf
RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine\&.
.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
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SS "subroutine dpoequ (integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)"
.PP
\fBDPOEQU\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A and reduce its condition number
(with respect to the two-norm)\&. S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal\&. This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
The N-by-N symmetric positive definite matrix whose scaling
factors are to be computed\&. Only the diagonal elements of A
are referenced\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is DOUBLE PRECISION array, dimension (N)
If INFO = 0, S contains the scale factors for A\&.
.fi
.PP
.br
\fISCOND\fP
.PP
.nf
SCOND is DOUBLE PRECISION
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i)\&. If SCOND >= 0\&.1 and AMAX is neither too
large nor too small, it is not worth scaling by S\&.
.fi
.PP
.br
\fIAMAX\fP
.PP
.nf
AMAX is DOUBLE PRECISION
Absolute value of largest matrix element\&. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled\&.
.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 i-th diagonal element is nonpositive\&.
.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 dpoequb (integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)"
.PP
\fBDPOEQUB\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOEQUB computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A and reduce its condition number
(with respect to the two-norm)\&. S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal\&. This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings\&.
This routine differs from DPOEQU by restricting the scaling factors
to a power of the radix\&. Barring over- and underflow, scaling by
these factors introduces no additional rounding errors\&. However, the
scaled diagonal entries are no longer approximately 1 but lie
between sqrt(radix) and 1/sqrt(radix)\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
The N-by-N symmetric positive definite matrix whose scaling
factors are to be computed\&. Only the diagonal elements of A
are referenced\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is DOUBLE PRECISION array, dimension (N)
If INFO = 0, S contains the scale factors for A\&.
.fi
.PP
.br
\fISCOND\fP
.PP
.nf
SCOND is DOUBLE PRECISION
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i)\&. If SCOND >= 0\&.1 and AMAX is neither too
large nor too small, it is not worth scaling by S\&.
.fi
.PP
.br
\fIAMAX\fP
.PP
.nf
AMAX is DOUBLE PRECISION
Absolute value of largest matrix element\&. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled\&.
.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 i-th diagonal element is nonpositive\&.
.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 dporfs (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, 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
\fBDPORFS\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPORFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite,
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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAF\fP
.PP
.nf
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDAF\fP
.PP
.nf
LDAF is INTEGER
The leading dimension of the array AF\&. LDAF >= max(1,N)\&.
.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 DPOTRS\&.
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
.SS "subroutine dporfsx (character UPLO, character EQUED, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)"
.PP
\fBDPORFSX\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPORFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive
definite, and provides error bounds and backward error estimates
for the solution\&. In addition to normwise error bound, the code
provides maximum componentwise error bound if possible\&. See
comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
error bounds\&.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED and S
below\&. In this case, the solution and error bounds returned are
for the original unequilibrated system\&.
.fi
.PP
.PP
.nf
Some optional parameters are bundled in the PARAMS array\&. These
settings determine how refinement is performed, but often the
defaults are acceptable\&. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument\&.
.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
\fIEQUED\fP
.PP
.nf
EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine\&. This is needed to compute
the solution and error bounds correctly\&.
= 'N': No equilibration
= 'Y': Both row and column equilibration, i\&.e\&., A has been
replaced by diag(S) * A * diag(S)\&.
The right hand side B has been changed accordingly\&.
.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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIAF\fP
.PP
.nf
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.fi
.PP
.br
\fILDAF\fP
.PP
.nf
LDAF is INTEGER
The leading dimension of the array AF\&. LDAF >= max(1,N)\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A\&. If EQUED = 'Y', A is multiplied on
the left and right by diag(S)\&. 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\&. If S is output, each
element of S is a power of the radix\&. If S is input, each element
of S should be a power of the radix to ensure a reliable solution
and error estimates\&. Scaling by powers of the radix does not cause
rounding errors unless the result underflows or overflows\&.
Rounding errors during scaling lead to refining with a matrix that
is not equivalent to the input matrix, producing error estimates
that may not be reliable\&.
.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 DGETRS\&.
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
\fIRCOND\fP
.PP
.nf
RCOND is DOUBLE PRECISION
Reciprocal scaled condition number\&. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done)\&. If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision\&. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned\&.
.fi
.PP
.br
\fIBERR\fP
.PP
.nf
BERR is DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error\&. This is 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
\fIN_ERR_BNDS\fP
.PP
.nf
N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise)\&. See ERR_BNDS_NORM and
ERR_BNDS_COMP below\&.
.fi
.PP
.br
\fIERR_BNDS_NORM\fP
.PP
.nf
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below\&. There currently are up to three pieces of information
returned\&.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side\&.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean\&. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon')\&.
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon')\&. This error bound should only
be trusted if the previous boolean is true\&.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number\&. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is 'guaranteed'\&. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z\&.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1\&.
See Lapack Working Note 165 for further details and extra
cautions\&.
.fi
.PP
.br
\fIERR_BNDS_COMP\fP
.PP
.nf
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below\&. There currently are up to three
pieces of information returned for each right-hand side\&. If
componentwise accuracy is not requested (PARAMS(3) = 0\&.0), then
ERR_BNDS_COMP is not accessed\&. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned\&.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side\&.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 'Trust/don't trust' boolean\&. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon')\&.
err = 2 'Guaranteed' error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon')\&. This error bound should only
be trusted if the previous boolean is true\&.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number\&. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is 'guaranteed'\&. These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z\&.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1\&.
See Lapack Working Note 165 for further details and extra
cautions\&.
.fi
.PP
.br
\fINPARAMS\fP
.PP
.nf
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS\&. If <= 0, the
PARAMS array is never referenced and default values are used\&.
.fi
.PP
.br
\fIPARAMS\fP
.PP
.nf
PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
Specifies algorithm parameters\&. If an entry is < 0\&.0, then
that entry will be filled with default value used for that
parameter\&. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters\&.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not\&.
Default: 1\&.0D+0
= 0\&.0: No refinement is performed, and no error bounds are
computed\&.
= 1\&.0: Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION\&.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement\&.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU\&. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy\&.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm\&. Positive
is true, 0\&.0 is false\&.
Default: 1\&.0 (attempt componentwise convergence)
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (4*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\&. The solution to every right-hand side is
guaranteed\&.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero\&. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed\&. RCOND = 0
is returned\&.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed\&. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported\&. If a small
componentwise error is not requested (PARAMS(3) = 0\&.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0\&.0)\&. By default (PARAMS(3) = 1\&.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0\&.0 or
ERR_BNDS_COMP(J,1) = 0\&.0)\&. See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1)\&. To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP\&.
.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 dpotf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)"
.PP
\fBDPOTF2\fP computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm)\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOTF2 computes the Cholesky factorization of a real symmetric
positive definite matrix A\&.
The factorization has the form
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 lower triangular\&.
This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored\&.
= 'U': Upper triangular
= 'L': Lower triangular
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 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 INFO = 0, 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
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k is not
positive definite, and the factorization could not be
completed\&.
.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 dpotrf (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)"
.PP
\fBDPOTRF\fP \fBDPOTRF\fP VARIANT: top-looking block version of the algorithm, calling Level 3 BLAS\&.
.PP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A\&.
The factorization has the form
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 lower triangular\&.
This is the block version of the algorithm, calling Level 3 BLAS\&.
.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
\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 INFO = 0, 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
\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 minor of order i is not
positive definite, and the factorization could not be
completed\&.
.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
\fBPurpose:\fP
.PP
.nf
DPOTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A\&.
The factorization has the form
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 lower triangular\&.
This is the top-looking block version of the algorithm, calling Level 3 BLAS\&.
.fi
.PP
.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
\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\&.
.fi
.PP
.PP
.nf
On exit, if INFO = 0, 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
\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 minor of order i is not
positive definite, and the factorization could not be
completed\&.
.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
\fBDate\fP
.RS 4
December 2016
.RE
.PP
.SS "recursive subroutine dpotrf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)"
.PP
\fBDPOTRF2\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOTRF2 computes the Cholesky factorization of a real symmetric
positive definite matrix A using the recursive algorithm\&.
The factorization has the form
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 lower triangular\&.
This is the recursive version of the algorithm\&. It divides
the matrix into four submatrices:
[ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
A = [ -----|----- ] with n1 = n/2
[ A21 | A22 ] n2 = n-n1
The subroutine calls itself to factor A11\&. Update and scale A21
or A12, update A22 then calls itself to factor A22\&.
.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
\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 INFO = 0, 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
\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 minor of order i is not
positive definite, and the factorization could not be
completed\&.
.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 dpotri (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)"
.PP
\fBDPOTRI\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOTRI computes the inverse of a real symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by DPOTRF\&.
.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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, as computed by
DPOTRF\&.
On exit, the upper or lower triangle of the (symmetric)
inverse of A, overwriting the input factor U or L\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,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
> 0: if INFO = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be 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 dpotrs (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)"
.PP
\fBDPOTRS\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DPOTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by DPOTRF\&.
.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 matrix B\&. NRHS >= 0\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF\&.
.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)
On entry, the right hand side matrix B\&.
On exit, the solution matrix X\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,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
\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\&.