## table of contents

realGBsolve(3) | LAPACK | realGBsolve(3) |

# NAME¶

realGBsolve - real

# SYNOPSIS¶

## Functions¶

subroutine **sgbsv** (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
INFO)

** SGBSV computes the solution to system of linear equations A * X = B for GB
matrices** (simple driver) subroutine **sgbsvx** (FACT, TRANS, N, KL,
KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND,
FERR, BERR, WORK, IWORK, INFO)

** SGBSVX computes the solution to system of linear equations A * X = B for
GB matrices** subroutine **sgbsvxx** (FACT, TRANS, N, KL, KU, NRHS, AB,
LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)

** SGBSVXX computes the solution to system of linear equations A * X = B for
GB matrices**

# Detailed Description¶

This is the group of real solve driver functions for GB matrices

# Function Documentation¶

## subroutine sgbsv (integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** SGBSV computes the solution to system of linear equations A *
X = B for GB matrices** (simple driver)

**Purpose:**

SGBSV computes the solution to a real system of linear equations

A * X = B, where A is a band matrix of order N with KL subdiagonals

and KU superdiagonals, and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is

used to factor A as A = L * U, where L is a product of permutation

and unit lower triangular matrices with KL subdiagonals, and U is

upper triangular with KL+KU superdiagonals. The factored form of A

is then used to solve the system of equations A * X = B.

**Parameters**

*N*

N is INTEGER

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

matrix A. N >= 0.

*KL*

KL is INTEGER

The number of subdiagonals within the band of A. KL >= 0.

*KU*

KU is INTEGER

The number of superdiagonals within the band of A. KU >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows KL+1 to

2*KL+KU+1; rows 1 to KL of the array need not be set.

The j-th column of A is stored in the j-th column of the

array AB as follows:

AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)

On exit, details of the factorization: U is stored as an

upper triangular band matrix with KL+KU superdiagonals in

rows 1 to KL+KU+1, and the multipliers used during the

factorization are stored in rows KL+KU+2 to 2*KL+KU+1.

See below for further details.

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= 2*KL+KU+1.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices that define the permutation matrix P;

row i of the matrix was interchanged with row IPIV(i).

*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, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and the solution has not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The band storage scheme is illustrated by the following example, when

M = N = 6, KL = 2, KU = 1:

On entry: On exit:

* * * + + + * * * u14 u25 u36

* * + + + + * * u13 u24 u35 u46

* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56

a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *

a31 a42 a53 a64 * * m31 m42 m53 m64 * *

Array elements marked * are not used by the routine; elements marked

+ need not be set on entry, but are required by the routine to store

elements of U because of fill-in resulting from the row interchanges.

## subroutine sgbsvx (character FACT, character TRANS, integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SGBSVX computes the solution to system of linear equations A *
X = B for GB matrices**

**Purpose:**

SGBSVX uses the LU factorization to compute the solution to a real

system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

where A is a band matrix of order N with KL subdiagonals and KU

superdiagonals, 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 by this subroutine:

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

the system:

TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B

TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B

TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')

or diag(C)*B (if TRANS = 'T' or 'C').

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

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

A = L * U,

where L is a product of permutation and unit lower triangular

matrices with KL subdiagonals, and U is upper triangular with

KL+KU superdiagonals.

3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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, AFB and IPIV contain the factored form of

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

equilibrated with scaling factors given by R and C.

AB, AFB, and IPIV are not modified.

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

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

copied to AFB and factored.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations.

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Transpose)

*N*

N is INTEGER

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

matrix A. N >= 0.

*KL*

KL is INTEGER

The number of subdiagonals within the band of A. KL >= 0.

*KU*

KU is INTEGER

The number of superdiagonals within the band of A. KU >= 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.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows 1 to KL+KU+1.

The j-th column of A is stored in the j-th column of the

array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then A must have been

equilibrated by the scaling factors in R and/or C. AB is not

modified if FACT = 'F' or 'N', or if FACT = 'E' and

EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A

EQUED = 'C': A := A * diag(C)

EQUED = 'B': A := diag(R) * A * diag(C).

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KL+KU+1.

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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

contains details of the LU factorization of the band matrix

A, as computed by SGBTRF. U is stored as an upper triangular

band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,

and the multipliers used during the factorization are stored

in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is

the factored form of the equilibrated matrix A.

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

returns details of the LU factorization of A.

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

returns details of the LU factorization of the equilibrated

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

equilibrated matrix).

*LDAFB*

LDAFB is INTEGER

The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

*IPIV*

IPIV is INTEGER array, dimension (N)

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

contains the pivot indices from the factorization A = L*U

as computed by SGBTRF; row i of the matrix was interchanged

with row IPIV(i).

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

contains the pivot indices from the factorization A = L*U

of the original matrix A.

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

contains the pivot indices from the factorization A = L*U

of the equilibrated matrix A.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

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

= 'R': Row equilibration, i.e., A has been premultiplied by

diag(R).

= 'C': Column equilibration, i.e., A has been postmultiplied

by diag(C).

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

replaced by diag(R) * A * diag(C).

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

output argument.

*R*

R is REAL array, dimension (N)

The row scale factors for A. If EQUED = 'R' or 'B', A is

multiplied on the left by diag(R); if EQUED = 'N' or 'C', R

is not accessed. R is an input argument if FACT = 'F';

otherwise, R is an output argument. If FACT = 'F' and

EQUED = 'R' or 'B', each element of R must be positive.

*C*

C is REAL array, dimension (N)

The column scale factors for A. If EQUED = 'C' or 'B', A is

multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

is not accessed. C is an input argument if FACT = 'F';

otherwise, C is an output argument. If FACT = 'F' and

EQUED = 'C' or 'B', each element of C must be positive.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit,

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

if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by

diag(R)*B;

if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is

overwritten by diag(C)*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 A and B are

modified on exit if EQUED .ne. 'N', and the solution to the

equilibrated system is inv(diag(C))*X if TRANS = 'N' and

EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'

and EQUED = 'R' or 'B'.

*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)

On exit, WORK(1) contains the reciprocal pivot growth

factor norm(A)/norm(U). The "max absolute element" norm is

used. If WORK(1) 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, condition

estimator RCOND, and forward error bound FERR could be

unreliable. If factorization fails with 0<INFO<=N, then

WORK(1) contains the reciprocal pivot growth factor for the

leading INFO columns of A.

*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: U(i,i) 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+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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sgbsvxx (character FACT, character TRANS, integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real RPVGRW, 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)¶

** SGBSVXX computes the solution to system of linear equations A
* X = B for GB matrices**

**Purpose:**

SGBSVXX uses the LU factorization to compute the solution to a

real system of linear equations A * X = B, where A is an

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

If requested, both normwise and maximum componentwise error bounds

are returned. SGBSVXX 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.

SGBSVXX 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

SGBSVXX 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 SGBSVXX would itself produce.

**Description:**

The following steps are performed:

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

the system:

TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B

TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B

TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')

or diag(C)*B (if TRANS = 'T' or 'C').

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

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

A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular

matrix, and U is upper triangular.

3. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A.

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

equilibrated with scaling factors given by R and C.

A, AF, and IPIV 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.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate Transpose = Transpose)

*N*

N is INTEGER

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

matrix A. N >= 0.

*KL*

KL is INTEGER

The number of subdiagonals within the band of A. KL >= 0.

*KU*

KU is INTEGER

The number of superdiagonals within the band of A. KU >= 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.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows 1 to KL+KU+1.

The j-th column of A is stored in the j-th column of the

array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then AB must have been

equilibrated by the scaling factors in R and/or C. AB is not

modified if FACT = 'F' or 'N', or if FACT = 'E' and

EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A

EQUED = 'C': A := A * diag(C)

EQUED = 'B': A := diag(R) * A * diag(C).

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KL+KU+1.

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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

contains details of the LU factorization of the band matrix

A, as computed by SGBTRF. U is stored as an upper triangular

band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,

and the multipliers used during the factorization are stored

in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is

the factored form of the equilibrated matrix A.

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

returns the factors L and U from the factorization A = P*L*U

of the original matrix A.

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

returns the factors L and U from the factorization A = P*L*U

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

the form of the equilibrated matrix).

*LDAFB*

LDAFB is INTEGER

The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

*IPIV*

IPIV is INTEGER array, dimension (N)

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

contains the pivot indices from the factorization A = P*L*U

as computed by SGETRF; row i of the matrix was interchanged

with row IPIV(i).

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

contains the pivot indices from the factorization A = P*L*U

of the original matrix A.

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

contains the pivot indices from the factorization A = P*L*U

of the equilibrated matrix A.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

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

= 'R': Row equilibration, i.e., A has been premultiplied by

diag(R).

= 'C': Column equilibration, i.e., A has been postmultiplied

by diag(C).

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

replaced by diag(R) * A * diag(C).

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

output argument.

*R*

R is REAL array, dimension (N)

The row scale factors for A. If EQUED = 'R' or 'B', A is

multiplied on the left by diag(R); if EQUED = 'N' or 'C', R

is not accessed. R is an input argument if FACT = 'F';

otherwise, R is an output argument. If FACT = 'F' and

EQUED = 'R' or 'B', each element of R must be positive.

If R is output, each element of R is a power of the radix.

If R is input, each element of R 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.

*C*

C is REAL array, dimension (N)

The column scale factors for A. If EQUED = 'C' or 'B', A is

multiplied on the right by diag(C); if EQUED = 'N' or 'R', C

is not accessed. C is an input argument if FACT = 'F';

otherwise, C is an output argument. If FACT = 'F' and

EQUED = 'C' or 'B', each element of C must be positive.

If C is output, each element of C is a power of the radix.

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)

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

On exit,

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

if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by

diag(R)*B;

if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is

overwritten by diag(C)*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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or

inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

*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. In SGESVX, this quantity is

returned in WORK(1).

*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.

# Author¶

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