complexGBcomputational(3) LAPACK complexGBcomputational(3)

complexGBcomputational - complex


subroutine cgbbrd (VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO)
CGBBRD subroutine cgbcon (NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, RWORK, INFO)
CGBCON subroutine cgbequ (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
CGBEQU subroutine cgbequb (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
CGBEQUB subroutine cgbrfs (TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CGBRFS subroutine cgbrfsx (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, RWORK, INFO)
CGBRFSX subroutine cgbtf2 (M, N, KL, KU, AB, LDAB, IPIV, INFO)
CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. subroutine cgbtrf (M, N, KL, KU, AB, LDAB, IPIV, INFO)
CGBTRF subroutine cgbtrs (TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
CGBTRS subroutine cggbak (JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
CGGBAK subroutine cggbal (JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
CGGBAL subroutine cla_gbamv (TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX, BETA, Y, INCY)
CLA_GBAMV performs a matrix-vector operation to calculate error bounds. real function cla_gbrcond_c (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, C, CAPPLY, INFO, WORK, RWORK)
CLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices. real function cla_gbrcond_x (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, X, INFO, WORK, RWORK)
CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices. subroutine cla_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)
CLA_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 cla_gbrpvgrw (N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
CLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix. subroutine cungbr (VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGBR

This is the group of complex computational functions for GB matrices

CGBBRD

Purpose:

CGBBRD reduces a complex general m-by-n band matrix A to real upper
bidiagonal form B by a unitary transformation: Q**H * A * P = B.
The routine computes B, and optionally forms Q or P**H, or computes
Q**H*C for a given matrix C.

Parameters

VECT
VECT is CHARACTER*1
Specifies whether or not the matrices Q and P**H are to be
formed.
= 'N': do not form Q or P**H;
= 'Q': form Q only;
= 'P': form P**H 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 COMPLEX 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 COMPLEX array, dimension (LDQ,M)
If VECT = 'Q' or 'B', the m-by-m unitary 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 COMPLEX array, dimension (LDPT,N)
If VECT = 'P' or 'B', the n-by-n unitary 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 COMPLEX array, dimension (LDC,NCC)
On entry, an m-by-ncc matrix C.
On exit, C is overwritten by Q**H*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 COMPLEX array, dimension (max(M,N))

RWORK

RWORK is REAL array, dimension (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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 191 of file cgbbrd.f.

CGBCON

Purpose:

CGBCON estimates the reciprocal of the condition number of a complex
general band matrix A, in either the 1-norm or the infinity-norm,
using the LU factorization computed by CGBTRF.
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 COMPLEX array, dimension (LDAB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 145 of file cgbcon.f.

CGBEQU

Purpose:

CGBEQU 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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 152 of file cgbequ.f.

CGBEQUB

Purpose:

CGBEQUB 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 CGEEQU 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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 159 of file cgbequb.f.

CGBRFS

Purpose:

CGBRFS 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)

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 COMPLEX 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 COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 CGBTRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

B

B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CGBTRS.
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 COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 203 of file cgbrfs.f.

CGBRFSX

Purpose:

CGBRFSX 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)

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 COMPLEX 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 COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 CGETRF; 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 COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CGETRS.
On exit, the improved solution matrix X.

LDX

     LDX is INTEGER
The leading dimension of the array X.  LDX >= max(1,N).

RCOND

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

BERR

     BERR is REAL array, dimension (NRHS)
Componentwise relative backward error.  This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).

N_ERR_BNDS

     N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise).  See ERR_BNDS_NORM and
ERR_BNDS_COMP below.

ERR_BNDS_NORM

     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * slamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated normwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * slamch('Epsilon') to determine if the error
         estimate is "guaranteed". These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*A, where S scales each row by a power of the
         radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP

     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * slamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated componentwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * slamch('Epsilon') to determine if the error
         estimate is "guaranteed". These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*(A*diag(x)), where x is the solution for the
         current right-hand side and S scales each row of
         A*diag(x) by a power of the radix so all absolute row
         sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.

NPARAMS

     NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS.  If <= 0, the
PARAMS array is never referenced and default values are used.

PARAMS

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

WORK

WORK is COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL array, dimension (2*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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 435 of file cgbrfsx.f.

CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose:

CGBTF2 computes an LU factorization of a complex 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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when
M = N = 6, KL = 2, KU = 1:
On entry:                       On exit:
    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
Array elements marked * are not used by the routine; elements marked
+ need not be set on entry, but are required by the routine to store
elements of U, because of fill-in resulting from the row
interchanges.

Definition at line 144 of file cgbtf2.f.

CGBTRF

Purpose:

CGBTRF computes an LU factorization of a complex 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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when
M = N = 6, KL = 2, KU = 1:
On entry:                       On exit:
    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
Array elements marked * are not used by the routine; elements marked
+ need not be set on entry, but are required by the routine to store
elements of U because of fill-in resulting from the row interchanges.

Definition at line 143 of file cgbtrf.f.

CGBTRS

Purpose:

CGBTRS solves a system of linear equations
   A * X = B,  A**T * X = B,  or  A**H * X = B
with a general band matrix A using the LU factorization computed
by CGBTRF.

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)

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 COMPLEX array, dimension (LDAB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 136 of file cgbtrs.f.

CGGBAK

Purpose:

CGGBAK forms the right or left eigenvectors of a complex generalized
eigenvalue problem A*x = lambda*B*x, by backward transformation on
the computed eigenvectors of the balanced pair of matrices output by
CGGBAL.

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 CGGBAL.

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 CGGBAL.
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 CGGBAL.

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 CGGBAL.

M

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

V

V is COMPLEX array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by CTGEVC.
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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

See R.C. Ward, Balancing the generalized eigenvalue problem,
               SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 146 of file cggbak.f.

CGGBAL

Purpose:

CGGBAL balances a pair of general complex 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 COMPLEX 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 COMPLEX 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
  RSCALE(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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

See R.C. WARD, Balancing the generalized eigenvalue problem,
               SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 175 of file cggbal.f.

CLA_GBAMV performs a matrix-vector operation to calculate error bounds.

Purpose:

CLA_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 COMPLEX 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, LDAB 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 COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 184 of file cla_gbamv.f.

CLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.

Purpose:

CLA_GBRCOND_C Computes the infinity norm condition number of
op(A) * inv(diag(C)) where C is a REAL vector.

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 COMPLEX 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 COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 CGBTRF; row i of the matrix was interchanged
with row IPIV(i).

C

     C is REAL array, dimension (N)
The vector C in the formula op(A) * inv(diag(C)).

CAPPLY

     CAPPLY is LOGICAL
If .TRUE. then access the vector C in the formula above.

INFO

     INFO is INTEGER
  = 0:  Successful exit.
i > 0:  The ith argument is invalid.

WORK

     WORK is COMPLEX array, dimension (2*N).
Workspace.

RWORK

     RWORK is REAL array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 158 of file cla_gbrcond_c.f.

CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.

Purpose:

CLA_GBRCOND_X Computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX vector.

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 COMPLEX 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 COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 CGBTRF; row i of the matrix was interchanged
with row IPIV(i).

X

     X is COMPLEX array, dimension (N)
The vector X in the formula op(A) * diag(X).

INFO

     INFO is INTEGER
  = 0:  Successful exit.
i > 0:  The ith argument is invalid.

WORK

     WORK is COMPLEX array, dimension (2*N).
Workspace.

RWORK

     RWORK is REAL array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 151 of file cla_gbrcond_x.f.

CLA_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:

CLA_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 CGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only resonsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters

PREC_TYPE
     PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
     = 'S':  Single
     = 'D':  Double
     = 'I':  Indigenous
     = 'X' or 'E':  Extra

TRANS_TYPE

     TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
     = 'N':  No transpose
     = 'T':  Transpose
     = 'C':  Conjugate transpose

N

     N is INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.

KL

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

KU

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

NRHS

     NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.

AB

     AB is COMPLEX 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 COMPLEX array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by CGBTRF.

LDAFB

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

IPIV

     IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by CGBTRF; 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 COMPLEX 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 COMPLEX array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by CGBTRS.
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 CLA_LIN_BERR.

N_NORMS

     N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.

ERR_BNDS_NORM

     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * slamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated normwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * slamch('Epsilon') to determine if the error
         estimate is "guaranteed". These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*A, where S scales each row by a power of the
         radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP

     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * slamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.
err = 3  Reciprocal condition number: Estimated componentwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * slamch('Epsilon') to determine if the error
         estimate is "guaranteed". These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*(A*diag(x)), where x is the solution for the
         current right-hand side and S scales each row of
         A*diag(x) by a power of the radix so all absolute row
         sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

RES

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

AYB

     AYB is REAL array, dimension (N)
Workspace.

DY

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

Y_TAIL

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

RCOND

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

ITHRESH

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

RTHRESH

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

DZ_UB

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

IGNORE_CWISE

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 403 of file cla_gbrfsx_extended.f.

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

Purpose:

CLA_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 COMPLEX 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 COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by CGBTRF.  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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 115 of file cla_gbrpvgrw.f.

CUNGBR

Purpose:

CUNGBR generates one of the complex unitary matrices Q or P**H
determined by CGEBRD when reducing a complex matrix A to bidiagonal
form: A = Q * B * P**H.  Q and P**H 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 CUNGBR returns the first n
columns of Q, where m >= n >= k;
if m < k, Q = H(1) H(2) . . . H(m-1) and CUNGBR returns Q as an
M-by-M matrix.
If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H
is of order N:
if k < n, P**H = G(k) . . . G(2) G(1) and CUNGBR returns the first m
rows of P**H, where n >= m >= k;
if k >= n, P**H = G(n-1) . . . G(2) G(1) and CUNGBR returns P**H as
an N-by-N matrix.

Parameters

VECT
VECT is CHARACTER*1
Specifies whether the matrix Q or the matrix P**H is
required, as defined in the transformation applied by CGEBRD:
= 'Q':  generate Q;
= 'P':  generate P**H.

M

M is INTEGER
The number of rows of the matrix Q or P**H to be returned.
M >= 0.

N

N is INTEGER
The number of columns of the matrix Q or P**H 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 CGEBRD.
If VECT = 'P', the number of rows in the original K-by-N
matrix reduced by CGEBRD.
K >= 0.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by CGEBRD.
On exit, the M-by-N matrix Q or P**H.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= M.

TAU

TAU is COMPLEX 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**H, as
returned by CGEBRD in its array argument TAUQ or TAUP.

WORK

WORK is COMPLEX 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 156 of file cungbr.f.

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