## table of contents

realPOcomputational(3) | LAPACK | realPOcomputational(3) |

# NAME¶

realPOcomputational

# SYNOPSIS¶

## Functions¶

real function **sla_porcond** (UPLO, N, A, LDA, AF, LDAF,
CMODE, C, INFO, WORK, IWORK)

**SLA_PORCOND** estimates the Skeel condition number for a symmetric
positive-definite matrix. subroutine **sla_porfsx_extended** (PREC_TYPE,
UPLO, N, NRHS, A, LDA, AF, LDAF, COLEQU, C, B, LDB, Y, LDY, BERR_OUT,
N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH,
RTHRESH, DZ_UB, IGNORE_CWISE, INFO)

**SLA_PORFSX_EXTENDED** improves the computed solution to a system of
linear equations for symmetric or Hermitian positive-definite matrices by
performing extra-precise iterative refinement and provides error bounds and
backward error estimates for the solution. real function **sla_porpvgrw**
(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)

**SLA_PORPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
subroutine **spocon** (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)

**SPOCON** subroutine **spoequ** (N, A, LDA, S, SCOND, AMAX, INFO)

**SPOEQU** subroutine **spoequb** (N, A, LDA, S, SCOND, AMAX, INFO)

**SPOEQUB** subroutine **sporfs** (UPLO, N, NRHS, A, LDA, AF, LDAF, B,
LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

**SPORFS** subroutine **sporfsx** (UPLO, EQUED, N, NRHS, A, LDA, AF,
LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

**SPORFSX** subroutine **spotf2** (UPLO, N, A, LDA, INFO)

**SPOTF2** computes the Cholesky factorization of a symmetric/Hermitian
positive definite matrix (unblocked algorithm). subroutine **spotrf**
(UPLO, N, A, LDA, INFO)

**SPOTRF** recursive subroutine **spotrf2** (UPLO, N, A, LDA, INFO)

**SPOTRF2** subroutine **spotri** (UPLO, N, A, LDA, INFO)

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

**SPOTRS**

# Detailed Description¶

This is the group of real computational functions for PO matrices

# Function Documentation¶

## real function sla_porcond (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)¶

**SLA_PORCOND** estimates the Skeel condition number for a
symmetric positive-definite matrix.

**Purpose:**

SLA_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.

**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.

*A*

A is REAL 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 REAL 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 SPOTRF.

*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 REAL 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 REAL array, dimension (3*N).

Workspace.

*IWORK*

IWORK is INTEGER array, dimension (N).

Workspace.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, real, dimension( * ) RES, real, dimension(*) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)¶

**SLA_PORFSX_EXTENDED** improves the computed solution to a
system of linear equations for symmetric or Hermitian positive-definite
matrices by performing extra-precise iterative refinement and provides error
bounds and backward error estimates for the solution.

**Purpose:**

SLA_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 SPORFSX 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 resonsible for setting the second fields of

ERR_BNDS_NORM and ERR_BNDS_COMP.

**Parameters**

*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 REAL 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 REAL 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 SPOTRF.

*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 REAL 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 REAL 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 REAL array, dimension (LDY,NRHS)

On entry, the solution matrix X, as computed by SPOTRS.

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 REAL 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 SLA_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 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.

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 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.

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 REAL array, dimension (N)

Workspace to hold the intermediate residual.

*AYB*

AYB is REAL array, dimension (N)

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

*DY*

DY is REAL array, dimension (N)

Workspace to hold the intermediate solution.

*Y_TAIL*

Y_TAIL is REAL array, dimension (N)

Workspace to hold the trailing bits of the intermediate solution.

*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.

*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 REAL

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 REAL

Determines when to start considering componentwise convergence.

Componentwise convergence is only considered after each component

of the solution Y is stable, which we definte 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 SPOTRS had an illegal

value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## real function sla_porpvgrw (character*1 UPLO, integer NCOLS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( * ) WORK)¶

**SLA_PORPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

**Purpose:**

SLA_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.

**Parameters**

*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 REAL 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 REAL 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 SPOTRF.

*LDAF*

LDAF is INTEGER

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

*WORK*

WORK is REAL array, dimension (2*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spocon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SPOCON**

**Purpose:**

SPOCON 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 SPOTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the

condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

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

*A*

A is REAL 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 SPOTRF.

*LDA*

LDA is INTEGER

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

*ANORM*

ANORM is REAL

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

*RCOND*

RCOND is REAL

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 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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spoequ (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)¶

**SPOEQU**

**Purpose:**

SPOEQU 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.

**Parameters**

*N*

N is INTEGER

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

*A*

A is REAL 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 REAL array, dimension (N)

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

*SCOND*

SCOND is REAL

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 REAL

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spoequb (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)¶

**SPOEQUB**

**Purpose:**

SPOEQUB 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 SPOEQU 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).

**Parameters**

*N*

N is INTEGER

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

*A*

A is REAL 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 REAL array, dimension (N)

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

*SCOND*

SCOND is REAL

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 REAL

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sporfs (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SPORFS**

**Purpose:**

SPORFS 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.

**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 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)

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 REAL 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 SPOTRF.

*LDAF*

LDAF is INTEGER

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

*B*

B is REAL 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 REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SPOTRS.

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 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

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sporfsx (character UPLO, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SPORFSX**

**Purpose:**

SPORFSX 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.

**Parameters**

*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 REAL 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 REAL 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 SPOTRF.

*LDAF*

LDAF is INTEGER

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

*S*

S is REAL 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 REAL 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 REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGETRS.

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 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.

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spotf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)¶

**SPOTF2** computes the Cholesky factorization of a
symmetric/Hermitian positive definite matrix (unblocked algorithm).

**Purpose:**

SPOTF2 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.

**Parameters**

*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 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).

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spotrf (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)¶

**SPOTRF** **SPOTRF** VARIANT: top-looking block version of
the algorithm, calling Level 3 BLAS.

**Purpose:**

SPOTRF 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.

**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 order of the matrix A. N >= 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).

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Purpose:**

SPOTRF 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.

**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 order of the matrix A. N >= 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).

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## recursive subroutine spotrf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)¶

**SPOTRF2**

**Purpose:**

SPOTRF2 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 call itself to factor A22.

**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 order of the matrix A. N >= 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).

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spotri (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)¶

**SPOTRI**

**Purpose:**

SPOTRI 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 SPOTRF.

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

*A*

A is REAL 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

SPOTRF.

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

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

**SPOTRS**

**Purpose:**

SPOTRS 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 SPOTRF.

**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 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)

The triangular factor U or L from the Cholesky factorization

A = U**T*U or A = L*L**T, as computed by SPOTRF.

*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 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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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