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.

UPLO

          UPLO is CHARACTER*1


       = 'U':  Upper triangle of A is stored;


       = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


     The number of linear equations, i.e., the order of the


     matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


     On entry, the N-by-N matrix A.

LDA

          LDA is INTEGER


     The leading dimension of the array A.  LDA >= max(1,N).

AF

          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.

LDAF

          LDAF is INTEGER


     The leading dimension of the array AF.  LDAF >= max(1,N).

CMODE

          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)

C

          C is DOUBLE PRECISION array, dimension (N)


     The vector C in the formula op(A) * op2(C).

INFO

          INFO is INTEGER


       = 0:  Successful exit.


     i > 0:  The ith argument is invalid.

WORK

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


     Workspace.

IWORK

          IWORK is INTEGER array, dimension (N).


     Workspace.

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 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.

PREC_TYPE

          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

UPLO

          UPLO is CHARACTER*1


       = 'U':  Upper triangle of A is stored;


       = 'L':  Lower triangle of A is stored.

N

          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


     matrix B.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


     On entry, the N-by-N matrix A.

LDA

          LDA is INTEGER


     The leading dimension of the array A.  LDA >= max(1,N).

AF

          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.

LDAF

          LDAF is INTEGER


     The leading dimension of the array AF.  LDAF >= max(1,N).

COLEQU

          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.

C

          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.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


     The right-hand-side matrix B.

LDB

          LDB is INTEGER


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

Y

          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.

LDY

          LDY is INTEGER


     The leading dimension of the array Y.  LDY >= max(1,N).

BERR_OUT

          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.

N_NORMS

          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.

ERR_BNDS_NORM

          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.

ERR_BNDS_COMP

          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.

RES

          RES is DOUBLE PRECISION array, dimension (N)


     Workspace to hold the intermediate residual.

AYB

          AYB is DOUBLE PRECISION array, dimension (N)


     Workspace. This can be the same workspace passed for Y_TAIL.

DY

          DY is DOUBLE PRECISION array, dimension (N)


     Workspace to hold the intermediate solution.

Y_TAIL

          Y_TAIL is DOUBLE PRECISION array, dimension (N)


     Workspace to hold the trailing bits of the intermediate solution.

RCOND

          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.

ITHRESH

          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.

RTHRESH

          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.

DZ_UB

          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.

IGNORE_CWISE

          IGNORE_CWISE is LOGICAL


     If .TRUE. then ignore componentwise convergence. Default value


     is .FALSE..

INFO

          INFO is INTEGER


       = 0:  Successful exit.


       < 0:  if INFO = -i, the ith argument to DPOTRS had an illegal


             value

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 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.

UPLO

          UPLO is CHARACTER*1


       = 'U':  Upper triangle of A is stored;


       = 'L':  Lower triangle of A is stored.

NCOLS

          NCOLS is INTEGER


     The number of columns of the matrix A. NCOLS >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


     On entry, the N-by-N matrix A.

LDA

          LDA is INTEGER


     The leading dimension of the array A.  LDA >= max(1,N).

AF

          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.

LDAF

          LDAF is INTEGER


     The leading dimension of the array AF.  LDAF >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)

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 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))).

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

ANORM

          ANORM is DOUBLE PRECISION


          The 1-norm (or infinity-norm) of the symmetric matrix A.

RCOND

          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.

WORK

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

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

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 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.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

S

          S is DOUBLE PRECISION array, dimension (N)


          If INFO = 0, S contains the scale factors for A.

SCOND

          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.

AMAX

          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.

INFO

          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.

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 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).

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

S

          S is DOUBLE PRECISION array, dimension (N)


          If INFO = 0, S contains the scale factors for A.

SCOND

          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.

AMAX

          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.

INFO

          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.

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 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          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.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          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.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          The right hand side matrix B.

LDB

          LDB is INTEGER


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

X

          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.

LDX

          LDX is INTEGER


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

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

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

  ITMAX is the maximum number of steps of iterative refinement.

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    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.

     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.

UPLO

          UPLO is CHARACTER*1


       = 'U':  Upper triangle of A is stored;


       = 'L':  Lower triangle of A is stored.

EQUED

          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.

N

          N is INTEGER


     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.

A

          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.

LDA

          LDA is INTEGER


     The leading dimension of the array A.  LDA >= max(1,N).

AF

          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.

LDAF

          LDAF is INTEGER


     The leading dimension of the array AF.  LDAF >= max(1,N).

S

          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.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


     The right hand side matrix B.

LDB

          LDB is INTEGER


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

X

          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.

LDX

          LDX is INTEGER


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

RCOND

          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.

BERR

          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).

N_ERR_BNDS

          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.

ERR_BNDS_NORM

          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.

ERR_BNDS_COMP

          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.

NPARAMS

          NPARAMS is INTEGER


     Specifies the number of parameters set in PARAMS.  If <= 0, the


     PARAMS array is never referenced and default values are used.

PARAMS

          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)

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          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.

Univ. of Tennessee

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NAG Ltd.

 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.

UPLO

          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

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          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.

Univ. of Tennessee

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Univ. of Colorado Denver

NAG Ltd.

 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          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 matrix B.  NRHS >= 0.

A

          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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          On entry, the right hand side matrix B.


          On exit, the solution matrix X.

LDB

          LDB is INTEGER


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

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.