## table of contents

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

# NAME¶

realGBcomputational# 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.

**Date:**

## 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.

**Date:**

## 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.

**Date:**

## 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.

**Date:**

## 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.

**Date:**

## 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 DGBTRF. 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 .LT. 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 .LE. 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 .LT. 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not. Default: 1.0 = 0.0 : No refinement is performed, and no error bounds are computed. = 1.0 : Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. (other values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)

*WORK*

WORK is REAL array, dimension (4*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO = -i, the i-th argument had an illegal value > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right- hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine 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.

**Date:**

**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.

**Date:**

**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.

**Date:**

## 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.

**Date:**

**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.

**Date:**

**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.

**Date:**

## 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.

**Date:**

## 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', '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 .LT. 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.

*RES*

RES is REAL array, dimension (N) Workspace to hold the intermediate residual.

*AYB*

AYB is REAL array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL.

*DY*

DY is REAL array, dimension (N) Workspace to hold the intermediate solution.

*Y_TAIL*

Y_TAIL is REAL array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.

*RCOND*

RCOND is REAL Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned.

*ITHRESH*

ITHRESH is INTEGER The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

*RTHRESH*

RTHRESH is REAL Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.

*DZ_UB*

DZ_UB is REAL Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we definte as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.

*IGNORE_CWISE*

IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

*INFO*

INFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to SGBTRS had an illegal value

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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.

**Date:**

## 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.

**Date:**

# Author¶

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