## table of contents

doublePOsolve(3) | LAPACK | doublePOsolve(3) |

# NAME¶

doublePOsolve

# SYNOPSIS¶

## Functions¶

subroutine **dposv** (UPLO, N, NRHS, A, LDA, B, LDB, INFO)

** DPOSV computes the solution to system of linear equations A * X = B for PO
matrices** subroutine **dposvx** (FACT, UPLO, N, NRHS, A, LDA, AF,
LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)

** DPOSVX computes the solution to system of linear equations A * X = B for
PO matrices** subroutine **dposvxx** (FACT, UPLO, N, NRHS, A, LDA, AF,
LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

** DPOSVXX computes the solution to system of linear equations A * X = B for
PO matrices** subroutine **dsposv** (UPLO, N, NRHS, A, LDA, B, LDB, X,
LDX, WORK, SWORK, ITER, INFO)

** DSPOSV computes the solution to system of linear equations A * X = B for
PO matrices**

# Detailed Description¶

This is the group of double solve driver functions for PO matrices

# Function Documentation¶

## subroutine dposv (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** DPOSV computes the solution to system of linear equations A *
X = B for PO matrices**

**Purpose:**

DPOSV 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.

**Parameters**

*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 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).

*B*

B is DOUBLE PRECISION 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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dposvx (character FACT, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, character EQUED, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** DPOSVX computes the solution to system of linear equations A *
X = B for PO matrices**

**Purpose:**

DPOSVX 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.

**Description:**

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.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION

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 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

> 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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dposvxx (character FACT, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, character EQUED, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision RPVGRW, 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)¶

** DPOSVXX computes the solution to system of linear equations A
* X = B for PO matrices**

**Purpose:**

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

to compute the solution to a double precision 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. DPOSVXX 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.

DPOSVXX 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

DPOSVXX 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 DPOSVXX would itself produce.

**Description:**

The following steps are performed:

1. If FACT = 'E', double precision 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.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.

*RPVGRW*

RPVGRW is DOUBLE PRECISION

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 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 extra-precise refinement algorithm.

(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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dsposv (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( n, * ) WORK, real, dimension( * ) SWORK, integer ITER, integer INFO)¶

** DSPOSV computes the solution to system of linear equations A *
X = B for PO matrices**

**Purpose:**

DSPOSV computes the solution to a real system of linear equations

A * X = B,

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

are N-by-NRHS matrices.

DSPOSV first attempts to factorize the matrix in SINGLE PRECISION

and use this factorization within an iterative refinement procedure

to produce a solution with DOUBLE PRECISION normwise backward error

quality (see below). If the approach fails the method switches to a

DOUBLE PRECISION factorization and solve.

The iterative refinement is not going to be a winning strategy if

the ratio SINGLE PRECISION performance over DOUBLE PRECISION

performance is too small. A reasonable strategy should take the

number of right-hand sides and the size of the matrix into account.

This might be done with a call to ILAENV in the future. Up to now, we

always try iterative refinement.

The iterative refinement process is stopped if

ITER > ITERMAX

or for all the RHS we have:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

o ITER is the number of the current iteration in the iterative

refinement process

o RNRM is the infinity-norm of the residual

o XNRM is the infinity-norm of the solution

o ANRM is the infinity-operator-norm of the matrix A

o EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00

respectively.

**Parameters**

*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 DOUBLE PRECISION array,

dimension (LDA,N)

On entry, the symmetric matrix A. If UPLO = 'U', the leading

N-by-N upper triangular part of A contains the upper

triangular part of the matrix A, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of A contains the lower

triangular part of the matrix A, and the strictly upper

triangular part of A is not referenced.

On exit, if iterative refinement has been successfully used

(INFO = 0 and ITER >= 0, see description below), then A is

unchanged, if double precision factorization has been used

(INFO = 0 and ITER < 0, see description below), then the

array A contains the factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T.

*LDA*

LDA is INTEGER

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

*B*

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

The N-by-NRHS 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)

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

*LDX*

LDX is INTEGER

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

*WORK*

WORK is DOUBLE PRECISION array, dimension (N,NRHS)

This array is used to hold the residual vectors.

*SWORK*

SWORK is REAL array, dimension (N*(N+NRHS))

This array is used to use the single precision matrix and the

right-hand sides or solutions in single precision.

*ITER*

ITER is INTEGER

< 0: iterative refinement has failed, double precision

factorization has been performed

-1 : the routine fell back to full precision for

implementation- or machine-specific reasons

-2 : narrowing the precision induced an overflow,

the routine fell back to full precision

-3 : failure of SPOTRF

-31: stop the iterative refinement after the 30th

iterations

> 0: iterative refinement has been successfully used.

Returns the number of iterations

*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 (DOUBLE

PRECISION) A is not positive definite, so the

factorization could not be completed, and the solution

has not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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