SPOSV computes the solution to a real system of linear equations


    A * X = B,


 where A is an N-by-N symmetric positive definite matrix and X and B


 are N-by-NRHS matrices.


 The Cholesky decomposition is used to factor A as


    A = U**T* U,  if UPLO = 'U', or


    A = L * L**T,  if UPLO = 'L',


 where U is an upper triangular matrix and L is a lower triangular


 matrix.  The factored form of A is then used to solve the system of


 equations A * X = B.

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.  NRHS >= 0.

A

          A is REAL 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).

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the N-by-NRHS right hand side matrix B.


          On exit, if INFO = 0, the N-by-NRHS 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


          > 0:  if INFO = i, the leading minor of order i of A is not


                positive definite, so the factorization could not be


                completed, and the solution has not been computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to


 compute the solution to a real system of linear equations


    A * X = B,


 where A is an N-by-N symmetric positive definite matrix and X and B


 are N-by-NRHS matrices.


 Error bounds on the solution and a condition estimate are also


 provided.

 The following steps are performed:


 1. If FACT = 'E', real scaling factors are computed to equilibrate


    the system:


       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B


    Whether or not the system will be equilibrated depends on the


    scaling of the matrix A, but if equilibration is used, A is


    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.


 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to


    factor the matrix A (after equilibration if FACT = 'E') as


       A = U**T* U,  if UPLO = 'U', or


       A = L * L**T,  if UPLO = 'L',


    where U is an upper triangular matrix and L is a lower triangular


    matrix.


 3. If the leading i-by-i principal minor is not positive definite,


    then the routine returns with INFO = i. Otherwise, the factored


    form of A is used to estimate the condition number of the matrix


    A.  If the reciprocal of the condition number is less than machine


    precision, INFO = N+1 is returned as a warning, but the routine


    still goes on to solve for X and compute error bounds as


    described below.


 4. The system of equations is solved for X using the factored form


    of A.


 5. Iterative refinement is applied to improve the computed solution


    matrix and calculate error bounds and backward error estimates


    for it.


 6. If equilibration was used, the matrix X is premultiplied by


    diag(S) so that it solves the original system before


    equilibration.

FACT

          FACT is CHARACTER*1


          Specifies whether or not the factored form of the matrix A is


          supplied on entry, and if not, whether the matrix A should be


          equilibrated before it is factored.


          = 'F':  On entry, AF contains the factored form of A.


                  If EQUED = 'Y', the matrix A has been equilibrated


                  with scaling factors given by S.  A and AF will not


                  be modified.


          = 'N':  The matrix A will be copied to AF and factored.


          = 'E':  The matrix A will be equilibrated if necessary, then


                  copied to AF and factored.

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 matrices B and X.  NRHS >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the symmetric matrix A, except if FACT = 'F' and


          EQUED = 'Y', then A must contain the equilibrated matrix


          diag(S)*A*diag(S).  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.  A is not modified if


          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by


          diag(S)*A*diag(S).

LDA

          LDA is INTEGER


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

AF

          AF is REAL array, dimension (LDAF,N)


          If FACT = 'F', then AF is an input argument and on entry


          contains the triangular factor U or L from the Cholesky


          factorization A = U**T*U or A = L*L**T, in the same storage


          format as A.  If EQUED .ne. 'N', then AF is the factored form


          of the equilibrated matrix diag(S)*A*diag(S).


          If FACT = 'N', then AF is an output argument and on exit


          returns the triangular factor U or L from the Cholesky


          factorization A = U**T*U or A = L*L**T of the original


          matrix A.


          If FACT = 'E', then AF is an output argument and on exit


          returns the triangular factor U or L from the Cholesky


          factorization A = U**T*U or A = L*L**T of the equilibrated


          matrix A (see the description of A for the form of the


          equilibrated matrix).

LDAF

          LDAF is INTEGER


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

EQUED

          EQUED is CHARACTER*1


          Specifies the form of equilibration that was done.


          = 'N':  No equilibration (always true if FACT = 'N').


          = 'Y':  Equilibration was done, i.e., A has been replaced by


                  diag(S) * A * diag(S).


          EQUED is an input argument if FACT = 'F'; otherwise, it is an


          output argument.

S

          S is REAL array, dimension (N)


          The scale factors for A; not accessed if EQUED = 'N'.  S is


          an input argument if FACT = 'F'; otherwise, S is an output


          argument.  If FACT = 'F' and EQUED = 'Y', each element of S


          must be positive.

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the N-by-NRHS right hand side matrix B.


          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',


          B is overwritten by diag(S) * B.

LDB

          LDB is INTEGER


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

X

          X is REAL array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to


          the original system of equations.  Note that if EQUED = 'Y',


          A and B are modified on exit, and the solution to the


          equilibrated system is inv(diag(S))*X.

LDX

          LDX is INTEGER


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

RCOND

          RCOND is REAL


          The estimate of the reciprocal condition number of the matrix


          A after equilibration (if done).  If RCOND is less than the


          machine precision (in particular, if RCOND = 0), the matrix


          is singular to working precision.  This condition is


          indicated by a return code of INFO > 0.

FERR

          FERR is REAL array, dimension (NRHS)


          The estimated forward error bound for each solution vector


          X(j) (the j-th column of the solution matrix X).


          If XTRUE is the true solution corresponding to X(j), FERR(j)


          is an estimated upper bound for the magnitude of the largest


          element in (X(j) - XTRUE) divided by the magnitude of the


          largest element in X(j).  The estimate is as reliable as


          the estimate for RCOND, and is almost always a slight


          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)


          The componentwise relative backward error of each solution


          vector X(j) (i.e., the smallest relative change in


          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is REAL 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


          > 0: if INFO = i, and i is


                <= N:  the leading minor of order i of A is


                       not positive definite, so the factorization


                       could not be completed, and the solution has not


                       been computed. RCOND = 0 is returned.


                = N+1: U is nonsingular, but RCOND is less than machine


                       precision, meaning that the matrix is singular


                       to working precision.  Nevertheless, the


                       solution and error bounds are computed because


                       there are a number of situations where the


                       computed solution can be more accurate than the


                       value of RCOND would suggest.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

    SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T


    to compute the solution to a real system of linear equations


    A * X = B, where A is an N-by-N symmetric positive definite matrix


    and X and B are N-by-NRHS matrices.


    If requested, both normwise and maximum componentwise error bounds


    are returned. SPOSVXX will return a solution with a tiny


    guaranteed error (O(eps) where eps is the working machine


    precision) unless the matrix is very ill-conditioned, in which


    case a warning is returned. Relevant condition numbers also are


    calculated and returned.


    SPOSVXX accepts user-provided factorizations and equilibration


    factors; see the definitions of the FACT and EQUED options.


    Solving with refinement and using a factorization from a previous


    SPOSVXX call will also produce a solution with either O(eps)


    errors or warnings, but we cannot make that claim for general


    user-provided factorizations and equilibration factors if they


    differ from what SPOSVXX would itself produce.

    The following steps are performed:


    1. If FACT = 'E', real scaling factors are computed to equilibrate


    the system:


      diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B


    Whether or not the system will be equilibrated depends on the


    scaling of the matrix A, but if equilibration is used, A is


    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.


    2. If FACT = 'N' or 'E', the Cholesky decomposition is used to


    factor the matrix A (after equilibration if FACT = 'E') as


       A = U**T* U,  if UPLO = 'U', or


       A = L * L**T,  if UPLO = 'L',


    where U is an upper triangular matrix and L is a lower triangular


    matrix.


    3. If the leading i-by-i principal minor is not positive definite,


    then the routine returns with INFO = i. Otherwise, the factored


    form of A is used to estimate the condition number of the matrix


    A (see argument RCOND).  If the reciprocal of the condition number


    is less than machine precision, the routine still goes on to solve


    for X and compute error bounds as described below.


    4. The system of equations is solved for X using the factored form


    of A.


    5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),


    the routine will use iterative refinement to try to get a small


    error and error bounds.  Refinement calculates the residual to at


    least twice the working precision.


    6. If equilibration was used, the matrix X is premultiplied by


    diag(S) so that it solves the original system before


    equilibration.

     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.

FACT

          FACT is CHARACTER*1


     Specifies whether or not the factored form of the matrix A is


     supplied on entry, and if not, whether the matrix A should be


     equilibrated before it is factored.


       = 'F':  On entry, AF contains the factored form of A.


               If EQUED is not 'N', the matrix A has been


               equilibrated with scaling factors given by S.


               A and AF are not modified.


       = 'N':  The matrix A will be copied to AF and factored.


       = 'E':  The matrix A will be equilibrated if necessary, then


               copied to AF and factored.

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 matrices B and X.  NRHS >= 0.

A

          A is REAL array, dimension (LDA,N)


     On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =


     'Y', then A must contain the equilibrated matrix


     diag(S)*A*diag(S).  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.  A is


     not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =


     'N' on exit.


     On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by


     diag(S)*A*diag(S).

LDA

          LDA is INTEGER


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

AF

          AF is REAL array, dimension (LDAF,N)


     If FACT = 'F', then AF is an input argument and on entry


     contains the triangular factor U or L from the Cholesky


     factorization A = U**T*U or A = L*L**T, in the same storage


     format as A.  If EQUED .ne. 'N', then AF is the factored


     form of the equilibrated matrix diag(S)*A*diag(S).


     If FACT = 'N', then AF is an output argument and on exit


     returns the triangular factor U or L from the Cholesky


     factorization A = U**T*U or A = L*L**T of the original


     matrix A.


     If FACT = 'E', then AF is an output argument and on exit


     returns the triangular factor U or L from the Cholesky


     factorization A = U**T*U or A = L*L**T of the equilibrated


     matrix A (see the description of A for the form of the


     equilibrated matrix).

LDAF

          LDAF is INTEGER


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

EQUED

          EQUED is CHARACTER*1


     Specifies the form of equilibration that was done.


       = 'N':  No equilibration (always true if FACT = 'N').


       = 'Y':  Both row and column equilibration, i.e., A has been


               replaced by diag(S) * A * diag(S).


     EQUED is an input argument if FACT = 'F'; otherwise, it is an


     output argument.

S

          S is REAL array, dimension (N)


     The row 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 REAL array, dimension (LDB,NRHS)


     On entry, the N-by-NRHS right hand side matrix B.


     On exit,


     if EQUED = 'N', B is not modified;


     if EQUED = 'Y', B is overwritten by diag(S)*B;

LDB

          LDB is INTEGER


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

X

          X is REAL array, dimension (LDX,NRHS)


     If INFO = 0, the N-by-NRHS solution matrix X to the original


     system of equations.  Note that A and B are modified on exit if


     EQUED .ne. 'N', and the solution to the equilibrated system is


     inv(diag(S))*X.

LDX

          LDX is INTEGER


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

RCOND

          RCOND is REAL


     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.

RPVGRW

          RPVGRW is REAL


     Reciprocal pivot growth.  On exit, this contains the reciprocal


     pivot growth factor norm(A)/norm(U). The 'max absolute element'


     norm is used.  If this is much less than 1, then 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. If factorization fails with


     0<INFO<=N, then this contains the reciprocal pivot growth factor


     for the leading INFO columns of A.

BERR

          BERR is REAL 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 REAL 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.


     See Lapack Working Note 165 for further details and extra


     cautions.

ERR_BNDS_COMP

          ERR_BNDS_COMP is REAL 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.


     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 REAL 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: 1.0


            = 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 REAL 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

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.