LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
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 matrixvector 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 extraprecise 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 
This is the group of real computational functions for GB matrices
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
Download SGBBRD + dependencies [TGZ] [ZIP] [TXT]SGBBRD reduces a real general mbyn 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.
[in]  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. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  NCC  NCC is INTEGER The number of columns of the matrix C. NCC >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals of the matrix A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals of the matrix A. KU >= 0. 
[in,out]  AB  AB is REAL array, dimension (LDAB,N) On entry, the mbyn band matrix A, stored in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(ku+1+ij,j) = A(i,j) for max(1,jku)<=i<=min(m,j+kl). On exit, A is overwritten by values generated during the reduction. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array A. LDAB >= KL+KU+1. 
[out]  D  D is REAL array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B. 
[out]  E  E is REAL array, dimension (min(M,N)1) The superdiagonal elements of the bidiagonal matrix B. 
[out]  Q  Q is REAL array, dimension (LDQ,M) If VECT = 'Q' or 'B', the mbym orthogonal matrix Q. If VECT = 'N' or 'P', the array Q is not referenced. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise. 
[out]  PT  PT is REAL array, dimension (LDPT,N) If VECT = 'P' or 'B', the nbyn orthogonal matrix P'. If VECT = 'N' or 'Q', the array PT is not referenced. 
[in]  LDPT  LDPT is INTEGER The leading dimension of the array PT. LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise. 
[in,out]  C  C is REAL array, dimension (LDC,NCC) On entry, an mbyncc matrix C. On exit, C is overwritten by Q**T*C. C is not referenced if NCC = 0. 
[in]  LDC  LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0. 
[out]  WORK  WORK is REAL array, dimension (2*max(M,N)) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 187 of file sgbbrd.f.
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
Download SGBCON + dependencies [TGZ] [ZIP] [TXT]SGBCON estimates the reciprocal of the condition number of a real general band matrix A, in either the 1norm or the infinitynorm, 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)) ).
[in]  NORM  NORM is CHARACTER*1 Specifies whether the 1norm condition number or the infinitynorm condition number is required: = '1' or 'O': 1norm; = 'I': Infinitynorm. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  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. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. 
[in]  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). 
[in]  ANORM  ANORM is REAL If NORM = '1' or 'O', the 1norm of the original matrix A. If NORM = 'I', the infinitynorm of the original matrix A. 
[out]  RCOND  RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(norm(A) * norm(inv(A))). 
[out]  WORK  WORK is REAL array, dimension (3*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 146 of file sgbcon.f.
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
Download SGBEQU + dependencies [TGZ] [ZIP] [TXT]SGBEQU computes row and column scalings intended to equilibrate an MbyN 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.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  AB  AB is REAL array, dimension (LDAB,N) The band matrix A, stored in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(ku+1+ij,j) = A(i,j) for max(1,jku)<=i<=min(m,j+kl). 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. 
[out]  R  R is REAL array, dimension (M) If INFO = 0, or INFO > M, R contains the row scale factors for A. 
[out]  C  C is REAL array, dimension (N) If INFO = 0, C contains the column scale factors for A. 
[out]  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. 
[out]  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. 
[out]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, and i is <= M: the ith row of A is exactly zero > M: the (iM)th column of A is exactly zero 
Definition at line 153 of file sgbequ.f.
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
Download SGBEQUB + dependencies [TGZ] [ZIP] [TXT]SGBEQUB computes row and column scalings intended to equilibrate an MbyN 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. Baring over and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitured are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  AB  AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(KU+1+ij,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl) 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array A. LDAB >= max(1,M). 
[out]  R  R is REAL array, dimension (M) If INFO = 0 or INFO > M, R contains the row scale factors for A. 
[out]  C  C is REAL array, dimension (N) If INFO = 0, C contains the column scale factors for A. 
[out]  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. 
[out]  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. 
[out]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, and i is <= M: the ith row of A is exactly zero > M: the (iM)th column of A is exactly zero 
Definition at line 160 of file sgbequb.f.
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
Download SGBRFS + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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) 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 
[in]  AB  AB is REAL array, dimension (LDAB,N) The original band matrix A, stored in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(ku+1+ij,j) = A(i,j) for max(1,jku)<=i<=min(n,j+kl). 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. 
[in]  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. 
[in]  LDAFB  LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. 
[in]  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). 
[in]  B  B is REAL array, dimension (LDB,NRHS) The right hand side matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  FERR  FERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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. 
[out]  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). 
[out]  WORK  WORK is REAL array, dimension (3*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
ITMAX is the maximum number of steps of iterative refinement.
Definition at line 204 of file sgbrfs.f.
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
Download SGBRFSX + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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) 
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 
[in]  AB  AB is DOUBLE PRECISION array, dimension (LDAB,N) The original band matrix A, stored in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(ku+1+ij,j) = A(i,j) for max(1,jku)<=i<=min(n,j+kl). 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. 
[in]  AFB  AFB is DOUBLE PRECISION 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. 
[in]  LDAFB  LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. 
[in]  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). 
[in,out]  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. 
[in,out]  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. 
[in]  B  B is REAL array, dimension (LDB,NRHS) The right hand side matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  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. 
[out]  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). 
[in]  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. 
[out]  ERR_BNDS_NORM  ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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. 
[out]  ERR_BNDS_COMP  ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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 righthand 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 righthand 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 righthand 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. 
[in]  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. 
[in,out]  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 highernumbered 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 doubleprecision refinement algorithm, possibly with doubledsingle 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 doubleprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence) 
[out]  WORK  WORK is REAL array, dimension (4*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: Successful exit. The solution to every righthand side is guaranteed. < 0: If INFO = i, the ith 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 righthand 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 righthand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth righthand 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 righthand 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 righthand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 
Definition at line 437 of file sgbrfsx.f.
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.
Download SGBTF2 + dependencies [TGZ] [ZIP] [TXT]SGBTF2 computes an LU factorization of a real mbyn band matrix A using partial pivoting with row interchanges. This is the unblocked version of the algorithm, calling Level 2 BLAS.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in,out]  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 jth column of A is stored in the jth column of the array AB as follows: AB(kl+ku+1+ij,j) = A(i,j) for max(1,jku)<=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. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. 
[out]  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). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith 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. 
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 fillin resulting from the row interchanges.
Definition at line 146 of file sgbtf2.f.
subroutine sgbtrf  (  integer  M, 
integer  N,  
integer  KL,  
integer  KU,  
real, dimension( ldab, * )  AB,  
integer  LDAB,  
integer, dimension( * )  IPIV,  
integer  INFO  
) 
SGBTRF
Download SGBTRF + dependencies [TGZ] [ZIP] [TXT]SGBTRF computes an LU factorization of a real mbyn band matrix A using partial pivoting with row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in,out]  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 jth column of A is stored in the jth column of the array AB as follows: AB(kl+ku+1+ij,j) = A(i,j) for max(1,jku)<=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. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. 
[out]  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). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith 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. 
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 fillin resulting from the row interchanges.
Definition at line 145 of file sgbtrf.f.
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
Download SGBTRS + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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) 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  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. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1. 
[in]  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). 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 138 of file sgbtrs.f.
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
Download SGGBAK + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  SIDE  SIDE is CHARACTER*1 = 'R': V contains right eigenvectors; = 'L': V contains left eigenvectors. 
[in]  N  N is INTEGER The number of rows of the matrix V. N >= 0. 
[in]  ILO  ILO is INTEGER 
[in]  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. 
[in]  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. 
[in]  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. 
[in]  M  M is INTEGER The number of columns of the matrix V. M >= 0. 
[in,out]  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. 
[in]  LDV  LDV is INTEGER The leading dimension of the matrix V. LDV >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
See R.C. Ward, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141152.
Definition at line 147 of file sggbak.f.
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
Download SGGBAL + dependencies [TGZ] [ZIP] [TXT]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 1norm of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x.
[in]  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. 
[in]  N  N is INTEGER The order of the matrices A and B. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in,out]  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. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  ILO  ILO is INTEGER 
[out]  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,...,ILO1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N. 
[out]  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,...,ILO1 = 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 ILO1. 
[out]  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,...,ILO1 = 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 ILO1. 
[out]  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'. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. 
See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141152.
Definition at line 177 of file sggbal.f.
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 matrixvector operation to calculate error bounds.
Download SLA_GBAMV + dependencies [TGZ] [ZIP] [TXT]SLA_GBAMV performs one of the matrixvector 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 blockstructure 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.
[in]  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. 
[in]  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. 
[in]  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. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  ALPHA  ALPHA is REAL On entry, ALPHA specifies the scalar alpha. Unchanged on exit. 
[in]  AB  AB is REAL array of DIMENSION ( LDAB, n ) Before entry, the leading m by n part of the array AB must contain the matrix of coefficients. Unchanged on exit. 
[in]  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. 
[in]  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. 
[in]  INCX  INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 
[in]  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. 
[in,out]  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 nonzero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. 
[in]  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. 
Definition at line 185 of file sla_gbamv.f.
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.
Download SLA_GBRCOND + dependencies [TGZ] [ZIP] [TXT]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 infinitynorm condition number.
[in]  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) 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  AB  AB is REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(KU+1+ij,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl) 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. 
[in]  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. 
[in]  LDAFB  LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 
[in]  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). 
[in]  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) 
[in]  C  C is REAL array, dimension (N) The vector C in the formula op(A) * op2(C). 
[out]  INFO  INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid. 
[in]  WORK  WORK is REAL array, dimension (5*N). Workspace. 
[in]  IWORK  IWORK is INTEGER array, dimension (N). Workspace. 
Definition at line 168 of file sla_gbrcond.f.
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 extraprecise iterative refinement and provides error bounds and backward error estimates for the solution.
Download SLA_GBRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations by performing extraprecise 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.
[in]  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 
[in]  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 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0 
[in]  NRHS  NRHS is INTEGER The number of righthandsides, i.e., the number of columns of the matrix B. 
[in]  AB  AB is REAL array, dimension (LDAB,N) On entry, the NbyN matrix AB. 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= max(1,N). 
[in]  AFB  AFB is REAL array, dimension (LDAFB,N) The factors L and U from the factorization A = P*L*U as computed by SGBTRF. 
[in]  LDAFB  LDAFB is INTEGER The leading dimension of the array AF. LDAFB >= max(1,N). 
[in]  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). 
[in]  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. 
[in]  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. 
[in]  B  B is REAL array, dimension (LDB,NRHS) The righthandside matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDY  LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N). 
[out]  BERR_OUT  BERR_OUT is REAL array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for righthandside 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. 
[in]  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. 
[in,out]  ERR_BNDS_NORM  ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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. 
[in,out]  ERR_BNDS_COMP  ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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 righthand 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 righthand 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 righthand 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. 
[in]  RES  RES is REAL array, dimension (N) Workspace to hold the intermediate residual. 
[in]  AYB  AYB is REAL array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL. 
[in]  DY  DY is REAL array, dimension (N) Workspace to hold the intermediate solution. 
[in]  Y_TAIL  Y_TAIL is REAL array, dimension (N) Workspace to hold the trailing bits of the intermediate solution. 
[in]  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. 
[in]  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. 
[in]  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 illconditioned matrices. See LAWN 165 for more details. 
[in]  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. 
[in]  IGNORE_CWISE  IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 
[out]  INFO  INFO is INTEGER = 0: Successful exit. < 0: if INFO = i, the ith argument to SGBTRS had an illegal value 
Definition at line 407 of file sla_gbrfsx_extended.f.
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.
Download SLA_GBRPVGRW + dependencies [TGZ] [ZIP] [TXT]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.
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. 
[in]  KU  KU is INTEGER The number of superdiagonals within the band of A. KU >= 0. 
[in]  NCOLS  NCOLS is INTEGER The number of columns of the matrix A. NCOLS >= 0. 
[in]  AB  AB is REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The jth column of A is stored in the jth column of the array AB as follows: AB(KU+1+ij,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl) 
[in]  LDAB  LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. 
[in]  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. 
[in]  LDAFB  LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 
Definition at line 117 of file sla_gbrpvgrw.f.
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
Download SORGBR + dependencies [TGZ] [ZIP] [TXT]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 MbyK 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(m1) and SORGBR returns Q as an MbyM matrix. If VECT = 'P', A is assumed to have been a KbyN 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(n1) . . . G(2) G(1) and SORGBR returns P**T as an NbyN matrix.
[in]  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. 
[in]  M  M is INTEGER The number of rows of the matrix Q or P**T to be returned. M >= 0. 
[in]  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). 
[in]  K  K is INTEGER If VECT = 'Q', the number of columns in the original MbyK matrix reduced by SGEBRD. If VECT = 'P', the number of rows in the original KbyN matrix reduced by SGEBRD. K >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the vectors which define the elementary reflectors, as returned by SGEBRD. On exit, the MbyN matrix Q or P**T. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in]  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. 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 
[in]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 158 of file sorgbr.f.