## table of contents

realGBcomputational(3) | LAPACK | realGBcomputational(3) |

# NAME¶

realGBcomputational - real

# SYNOPSIS¶

## Functions¶

subroutine **sgbbrd** (VECT, M, N, NCC, KL, KU, AB, LDAB, D, E,
Q, LDQ, PT, LDPT, C, LDC, WORK, INFO)

**SGBBRD** subroutine **sgbcon** (NORM, N, KL, KU, AB, LDAB, IPIV,
ANORM, RCOND, WORK, IWORK, INFO)

**SGBCON** subroutine **sgbequ** (M, N, KL, KU, AB, LDAB, R, C, ROWCND,
COLCND, AMAX, INFO)

**SGBEQU** subroutine **sgbequb** (M, N, KL, KU, AB, LDAB, R, C, ROWCND,
COLCND, AMAX, INFO)

**SGBEQUB** subroutine **sgbrfs** (TRANS, N, KL, KU, NRHS, AB, LDAB,
AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

**SGBRFS** subroutine **sgbrfsx** (TRANS, EQUED, N, KL, KU, NRHS, AB,
LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

**SGBRFSX** subroutine **sgbtf2** (M, N, KL, KU, AB, LDAB, IPIV, INFO)

**SGBTF2** computes the LU factorization of a general band matrix using the
unblocked version of the algorithm. subroutine **sgbtrf** (M, N, KL, KU,
AB, LDAB, IPIV, INFO)

**SGBTRF** subroutine **sgbtrs** (TRANS, N, KL, KU, NRHS, AB, LDAB,
IPIV, B, LDB, INFO)

**SGBTRS** subroutine **sggbak** (JOB, SIDE, N, ILO, IHI, LSCALE,
RSCALE, M, V, LDV, INFO)

**SGGBAK** subroutine **sggbal** (JOB, N, A, LDA, B, LDB, ILO, IHI,
LSCALE, RSCALE, WORK, INFO)

**SGGBAL** subroutine **sla_gbamv** (TRANS, M, N, KL, KU, ALPHA, AB,
LDAB, X, INCX, BETA, Y, INCY)

**SLA_GBAMV** performs a matrix-vector operation to calculate error bounds.
real function **sla_gbrcond** (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB,
IPIV, CMODE, C, INFO, WORK, IWORK)

**SLA_GBRCOND** estimates the Skeel condition number for a general banded
matrix. subroutine **sla_gbrfsx_extended** (PREC_TYPE, TRANS_TYPE, N, KL,
KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, 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_GBRFSX_EXTENDED** improves the computed solution to a system of
linear equations for general banded matrices by performing extra-precise
iterative refinement and provides error bounds and backward error estimates
for the solution. real function **sla_gbrpvgrw** (N, KL, KU, NCOLS, AB,
LDAB, AFB, LDAFB)

**SLA_GBRPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a general banded matrix. subroutine **sorgbr** (VECT,
M, N, K, A, LDA, TAU, WORK, LWORK, INFO)

**SORGBR**

# Detailed Description¶

This is the group of real computational functions for GB matrices

# Function Documentation¶

## subroutine sgbbrd (character VECT, integer M, integer N, integer NCC, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldpt, * ) PT, integer LDPT, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)¶

**SGBBRD**

**Purpose:**

SGBBRD reduces a real general m-by-n band matrix A to upper

bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

The routine computes B, and optionally forms Q or P**T, or computes

Q**T*C for a given matrix C.

**Parameters**

*VECT*

VECT is CHARACTER*1

Specifies whether or not the matrices Q and P**T are to be

formed.

= 'N': do not form Q or P**T;

= 'Q': form Q only;

= 'P': form P**T only;

= 'B': form both.

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

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

*NCC*

NCC is INTEGER

The number of columns of the matrix C. NCC >= 0.

*KL*

KL is INTEGER

The number of subdiagonals of the matrix A. KL >= 0.

*KU*

KU is INTEGER

The number of superdiagonals of the matrix A. KU >= 0.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, the m-by-n band matrix A, stored 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(m,j+kl).

On exit, A is overwritten by values generated during the

reduction.

*LDAB*

LDAB is INTEGER

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

*D*

D is REAL array, dimension (min(M,N))

The diagonal elements of the bidiagonal matrix B.

*E*

E is REAL array, dimension (min(M,N)-1)

The superdiagonal elements of the bidiagonal matrix B.

*Q*

Q is REAL array, dimension (LDQ,M)

If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.

If VECT = 'N' or 'P', the array Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q.

LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.

*PT*

PT is REAL array, dimension (LDPT,N)

If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.

If VECT = 'N' or 'Q', the array PT is not referenced.

*LDPT*

LDPT is INTEGER

The leading dimension of the array PT.

LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.

*C*

C is REAL array, dimension (LDC,NCC)

On entry, an m-by-ncc matrix C.

On exit, C is overwritten by Q**T*C.

C is not referenced if NCC = 0.

*LDC*

LDC is INTEGER

The leading dimension of the array C.

LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.

*WORK*

WORK is REAL array, dimension (2*max(M,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.

## subroutine sgbcon (character NORM, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SGBCON**

**Purpose:**

SGBCON estimates the reciprocal of the condition number of a real

general band matrix A, in either the 1-norm or the infinity-norm,

using the LU factorization computed by SGBTRF.

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

condition number is computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) ).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= '1' or 'O': 1-norm;

= 'I': Infinity-norm.

*N*

N is INTEGER

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.

*AB*

AB is REAL array, dimension (LDAB,N)

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.

*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; for 1 <= i <= N, row i of the matrix was

interchanged with row IPIV(i).

*ANORM*

ANORM is REAL

If NORM = '1' or 'O', the 1-norm of the original matrix A.

If NORM = 'I', the infinity-norm of the original matrix A.

*RCOND*

RCOND is REAL

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(norm(A) * norm(inv(A))).

*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.

## subroutine sgbequ (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)¶

**SGBEQU**

**Purpose:**

SGBEQU computes row and column scalings intended to equilibrate an

M-by-N band matrix A and reduce its condition number. R returns the

row scale factors and C the column scale factors, chosen to try to

make the largest element in each row and column of the matrix B with

elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

R(i) and C(j) are restricted to be between SMLNUM = smallest safe

number and BIGNUM = largest safe number. Use of these scaling

factors is not guaranteed to reduce the condition number of A but

works well in practice.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns 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.

*AB*

AB is REAL array, dimension (LDAB,N)

The band matrix A, stored 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(m,j+kl).

*LDAB*

LDAB is INTEGER

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

*R*

R is REAL array, dimension (M)

If INFO = 0, or INFO > M, R contains the row scale factors

for A.

*C*

C is REAL array, dimension (N)

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

*ROWCND*

ROWCND is REAL

If INFO = 0 or INFO > M, ROWCND contains the ratio of the

smallest R(i) to the largest R(i). If ROWCND >= 0.1 and

AMAX is neither too large nor too small, it is not worth

scaling by R.

*COLCND*

COLCND is REAL

If INFO = 0, COLCND contains the ratio of the smallest

C(i) to the largest C(i). If COLCND >= 0.1, it is not

worth scaling by C.

*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, and i is

<= M: the i-th row of A is exactly zero

> M: the (i-M)-th column of A is exactly zero

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sgbequb (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)¶

**SGBEQUB**

**Purpose:**

SGBEQUB computes row and column scalings intended to equilibrate an

M-by-N matrix A and reduce its condition number. R returns the row

scale factors and C the column scale factors, chosen to try to make

the largest element in each row and column of the matrix B with

elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most

the radix.

R(i) and C(j) are restricted to be a power of the radix between

SMLNUM = smallest safe number and BIGNUM = largest safe number. Use

of these scaling factors is not guaranteed to reduce the condition

number of A but works well in practice.

This routine differs from SGEEQU 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 entries' magnitudes are no longer approximately 1 but lie

between sqrt(radix) and 1/sqrt(radix).

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns 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.

*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)

*LDAB*

LDAB is INTEGER

The leading dimension of the array A. LDAB >= max(1,M).

*R*

R is REAL array, dimension (M)

If INFO = 0 or INFO > M, R contains the row scale factors

for A.

*C*

C is REAL array, dimension (N)

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

*ROWCND*

ROWCND is REAL

If INFO = 0 or INFO > M, ROWCND contains the ratio of the

smallest R(i) to the largest R(i). If ROWCND >= 0.1 and

AMAX is neither too large nor too small, it is not worth

scaling by R.

*COLCND*

COLCND is REAL

If INFO = 0, COLCND contains the ratio of the smallest

C(i) to the largest C(i). If COLCND >= 0.1, it is not

worth scaling by C.

*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, and i is

<= M: the i-th row of A is exactly zero

> M: the (i-M)-th column of A is exactly zero

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sgbrfs (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, 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)¶

**SGBRFS**

**Purpose:**

SGBRFS improves the computed solution to a system of linear

equations when the coefficient matrix is banded, and provides

error bounds and backward error estimates for the solution.

**Parameters**

*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 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)

The original band matrix A, stored 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).

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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.

*LDAFB*

LDAFB is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from SGBTRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*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 SGBTRS.

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.

## subroutine sgbrfsx (character TRANS, character EQUED, integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, real, dimension( * ) R, real, dimension( * ) C, 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)¶

**SGBRFSX**

**Purpose:**

SGBRFSX improves the computed solution to a system of linear

equations 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, R

and C 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**

*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)

*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

= '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).

The right hand side B has been changed accordingly.

*N*

N is INTEGER

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)

The original band matrix A, stored 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).

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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.

*LDAFB*

LDAFB is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from SGETRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*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)

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.

## subroutine sgbtf2 (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, integer INFO)¶

**SGBTF2** computes the LU factorization of a general band
matrix using the unblocked version of the algorithm.

**Purpose:**

SGBTF2 computes an LU factorization of a real m-by-n band matrix A

using partial pivoting with row interchanges.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns 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.

*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(m,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 (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*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 division by zero will occur if it is used

to solve a system of equations.

**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 sgbtrf (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, integer INFO)¶

**SGBTRF**

**Purpose:**

SGBTRF computes an LU factorization of a real m-by-n band matrix A

using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns 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.

*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(m,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 (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*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 division by zero will occur if it is used

to solve a system of equations.

**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 sgbtrs (character TRANS, 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)¶

**SGBTRS**

**Purpose:**

SGBTRS solves a system of linear equations

A * X = B or A**T * X = B

with a general band matrix A using the LU factorization computed

by SGBTRF.

**Parameters**

*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**T* X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

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)

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.

*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; for 1 <= i <= N, row i of the matrix was

interchanged with row IPIV(i).

*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.

## subroutine sggbak (character JOB, character SIDE, integer N, integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, integer M, real, dimension( ldv, * ) V, integer LDV, integer INFO)¶

**SGGBAK**

**Purpose:**

SGGBAK forms the right or left eigenvectors of a real generalized

eigenvalue problem A*x = lambda*B*x, by backward transformation on

the computed eigenvectors of the balanced pair of matrices output by

SGGBAL.

**Parameters**

*JOB*

JOB is CHARACTER*1

Specifies the type of backward transformation required:

= 'N': do nothing, return immediately;

= 'P': do backward transformation for permutation only;

= 'S': do backward transformation for scaling only;

= 'B': do backward transformations for both permutation and

scaling.

JOB must be the same as the argument JOB supplied to SGGBAL.

*SIDE*

SIDE is CHARACTER*1

= 'R': V contains right eigenvectors;

= 'L': V contains left eigenvectors.

*N*

N is INTEGER

The number of rows of the matrix V. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

The integers ILO and IHI determined by SGGBAL.

1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*LSCALE*

LSCALE is REAL array, dimension (N)

Details of the permutations and/or scaling factors applied

to the left side of A and B, as returned by SGGBAL.

*RSCALE*

RSCALE is REAL array, dimension (N)

Details of the permutations and/or scaling factors applied

to the right side of A and B, as returned by SGGBAL.

*M*

M is INTEGER

The number of columns of the matrix V. M >= 0.

*V*

V is REAL array, dimension (LDV,M)

On entry, the matrix of right or left eigenvectors to be

transformed, as returned by STGEVC.

On exit, V is overwritten by the transformed eigenvectors.

*LDV*

LDV is INTEGER

The leading dimension of the matrix V. LDV >= 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.

**Further Details:**

See R.C. Ward, Balancing the generalized eigenvalue problem,

SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

## subroutine sggbal (character JOB, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real, dimension( * ) WORK, integer INFO)¶

**SGGBAL**

**Purpose:**

SGGBAL balances a pair of general real matrices (A,B). This

involves, first, permuting A and B by similarity transformations to

isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N

elements on the diagonal; and second, applying a diagonal similarity

transformation to rows and columns ILO to IHI to make the rows

and columns as close in norm as possible. Both steps are optional.

Balancing may reduce the 1-norm of the matrices, and improve the

accuracy of the computed eigenvalues and/or eigenvectors in the

generalized eigenvalue problem A*x = lambda*B*x.

**Parameters**

*JOB*

JOB is CHARACTER*1

Specifies the operations to be performed on A and B:

= 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0

and RSCALE(I) = 1.0 for i = 1,...,N.

= 'P': permute only;

= 'S': scale only;

= 'B': both permute and scale.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the input matrix A.

On exit, A is overwritten by the balanced matrix.

If JOB = 'N', A is not referenced.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB,N)

On entry, the input matrix B.

On exit, B is overwritten by the balanced matrix.

If JOB = 'N', B is not referenced.

*LDB*

LDB is INTEGER

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

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

ILO and IHI are set to integers such that on exit

A(i,j) = 0 and B(i,j) = 0 if i > j and

j = 1,...,ILO-1 or i = IHI+1,...,N.

If JOB = 'N' or 'S', ILO = 1 and IHI = N.

*LSCALE*

LSCALE is REAL array, dimension (N)

Details of the permutations and scaling factors applied

to the left side of A and B. If P(j) is the index of the

row interchanged with row j, and D(j)

is the scaling factor applied to row j, then

LSCALE(j) = P(j) for J = 1,...,ILO-1

= D(j) for J = ILO,...,IHI

= P(j) for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1,

then 1 to ILO-1.

*RSCALE*

RSCALE is REAL array, dimension (N)

Details of the permutations and scaling factors applied

to the right side of A and B. If P(j) is the index of the

column interchanged with column j, and D(j)

is the scaling factor applied to column j, then

LSCALE(j) = P(j) for J = 1,...,ILO-1

= D(j) for J = ILO,...,IHI

= P(j) for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1,

then 1 to ILO-1.

*WORK*

WORK is REAL array, dimension (lwork)

lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and

at least 1 when JOB = 'N' or 'P'.

*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.

**Further Details:**

See R.C. WARD, Balancing the generalized eigenvalue problem,

SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

## subroutine sla_gbamv (integer TRANS, integer M, integer N, integer KL, integer KU, real ALPHA, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)¶

**SLA_GBAMV** performs a matrix-vector operation to calculate
error bounds.

**Purpose:**

SLA_GBAMV performs one of the matrix-vector operations

y := alpha*abs(A)*abs(x) + beta*abs(y),

or y := alpha*abs(A)**T*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an

m by n matrix.

This function is primarily used in calculating error bounds.

To protect against underflow during evaluation, components in

the resulting vector are perturbed away from zero by (N+1)

times the underflow threshold. To prevent unnecessarily large

errors for block-structure embedded in general matrices,

"symbolically" zero components are not perturbed. A zero

entry is considered "symbolic" if all multiplications involved

in computing that entry have at least one zero multiplicand.

**Parameters**

*TRANS*

TRANS is INTEGER

On entry, TRANS specifies the operation to be performed as

follows:

BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)

BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)

BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)

Unchanged on exit.

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

Unchanged on exit.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

Unchanged on exit.

*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.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.

*AB*

AB is REAL array, dimension ( LDAB, n )

Before entry, the leading m by n part of the array AB must

contain the matrix of coefficients.

Unchanged on exit.

*LDAB*

LDAB is INTEGER

On entry, LDA specifies the first dimension of AB as declared

in the calling (sub) program. LDAB must be at least

max( 1, m ).

Unchanged on exit.

*X*

X is REAL array, dimension

( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'

and at least

( 1 + ( m - 1 )*abs( INCX ) ) otherwise.

Before entry, the incremented array X must contain the

vector x.

Unchanged on exit.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

Unchanged on exit.

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta. When BETA is

supplied as zero then Y need not be set on input.

Unchanged on exit.

*Y*

Y is REAL array, dimension

( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'

and at least

( 1 + ( n - 1 )*abs( INCY ) ) otherwise.

Before entry with BETA non-zero, the incremented array Y

must contain the vector y. On exit, Y is overwritten by the

updated vector y.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

Unchanged on exit.

Level 2 Blas routine.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## real function sla_gbrcond (character TRANS, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)¶

**SLA_GBRCOND** estimates the Skeel condition number for a
general banded matrix.

**Purpose:**

SLA_GBRCOND 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**

*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.

*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)

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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.

*LDAFB*

LDAFB is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

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

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

with row IPIV(i).

*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 (5*N).

Workspace.

*IWORK*

IWORK is INTEGER array, dimension (N).

Workspace.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sla_gbrfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer KL, integer KU, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV, 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_GBRFSX_EXTENDED** improves the computed solution to a
system of linear equations for general banded matrices by performing
extra-precise iterative refinement and provides error bounds and backward
error estimates for the solution.

**Purpose:**

SLA_GBRFSX_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 SGBRFSX 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

*TRANS_TYPE*

TRANS_TYPE is INTEGER

Specifies the transposition operation on A.

The value is defined by ILATRANS(T) where T is a CHARACTER and T

= 'N': No transpose

= 'T': Transpose

= 'C': Conjugate 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

matrix B.

*AB*

AB is REAL array, dimension (LDAB,N)

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

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= max(1,N).

*AFB*

AFB is REAL array, dimension (LDAFB,N)

The factors L and U from the factorization

A = P*L*U as computed by SGBTRF.

*LDAFB*

LDAFB is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

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

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

with row IPIV(i).

*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 SGBTRS.

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 define as the relative

change in each component being less than DZ_UB. The default value

is 0.25, requiring the first bit to be stable. See LAWN 165 for

more details.

*IGNORE_CWISE*

IGNORE_CWISE is LOGICAL

If .TRUE. then ignore componentwise convergence. Default value

is .FALSE..

*INFO*

INFO is INTEGER

= 0: Successful exit.

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

value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## real function sla_gbrpvgrw (integer N, integer KL, integer KU, integer NCOLS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB)¶

**SLA_GBRPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a general banded matrix.

**Purpose:**

SLA_GBRPVGRW 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**

*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.

*NCOLS*

NCOLS is INTEGER

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

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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.

*LDAFB*

LDAFB is INTEGER

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sorgbr (character VECT, integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SORGBR**

**Purpose:**

SORGBR generates one of the real orthogonal matrices Q or P**T

determined by SGEBRD when reducing a real matrix A to bidiagonal

form: A = Q * B * P**T. Q and P**T are defined as products of

elementary reflectors H(i) or G(i) respectively.

If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q

is of order M:

if m >= k, Q = H(1) H(2) . . . H(k) and SORGBR returns the first n

columns of Q, where m >= n >= k;

if m < k, Q = H(1) H(2) . . . H(m-1) and SORGBR returns Q as an

M-by-M matrix.

If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T

is of order N:

if k < n, P**T = G(k) . . . G(2) G(1) and SORGBR returns the first m

rows of P**T, where n >= m >= k;

if k >= n, P**T = G(n-1) . . . G(2) G(1) and SORGBR returns P**T as

an N-by-N matrix.

**Parameters**

*VECT*

VECT is CHARACTER*1

Specifies whether the matrix Q or the matrix P**T is

required, as defined in the transformation applied by SGEBRD:

= 'Q': generate Q;

= 'P': generate P**T.

*M*

M is INTEGER

The number of rows of the matrix Q or P**T to be returned.

M >= 0.

*N*

N is INTEGER

The number of columns of the matrix Q or P**T to be returned.

N >= 0.

If VECT = 'Q', M >= N >= min(M,K);

if VECT = 'P', N >= M >= min(N,K).

*K*

K is INTEGER

If VECT = 'Q', the number of columns in the original M-by-K

matrix reduced by SGEBRD.

If VECT = 'P', the number of rows in the original K-by-N

matrix reduced by SGEBRD.

K >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the vectors which define the elementary reflectors,

as returned by SGEBRD.

On exit, the M-by-N matrix Q or P**T.

*LDA*

LDA is INTEGER

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

*TAU*

TAU is REAL array, dimension

(min(M,K)) if VECT = 'Q'

(min(N,K)) if VECT = 'P'

TAU(i) must contain the scalar factor of the elementary

reflector H(i) or G(i), which determines Q or P**T, as

returned by SGEBRD in its array argument TAUQ or TAUP.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,min(M,N)).

For optimum performance LWORK >= min(M,N)*NB, where NB

is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*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.

# Author¶

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