# COMPLEX routines for general band matrix

## cgbbrd

```USAGE:
d, e, q, pt, info, ab, c = NumRu::Lapack.cgbbrd( vect, kl, ku, ab, c, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )

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

*  Arguments
*  =========
*
*  VECT    (input) CHARACTER*1
*          Specifies whether or not the matrices Q and P' are to be
*          formed.
*          = 'N': do not form Q or P';
*          = 'Q': form Q only;
*          = 'P': form P' only;
*          = 'B': form both.
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  NCC     (input) INTEGER
*          The number of columns of the matrix C.  NCC >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals of the matrix A. KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals of the matrix A. KU >= 0.
*
*  AB      (input/output) 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    (input) INTEGER
*          The leading dimension of the array A. LDAB >= KL+KU+1.
*
*  D       (output) REAL array, dimension (min(M,N))
*          The diagonal elements of the bidiagonal matrix B.
*
*  E       (output) REAL array, dimension (min(M,N)-1)
*          The superdiagonal elements of the bidiagonal matrix B.
*
*  Q       (output) 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     (input) INTEGER
*          The leading dimension of the array Q.
*          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
*
*  PT      (output) 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    (input) INTEGER
*          The leading dimension of the array PT.
*          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
*
*  C       (input/output) COMPLEX array, dimension (LDC,NCC)
*          On entry, an m-by-ncc matrix C.
*          On exit, C is overwritten by Q'*C.
*          C is not referenced if NCC = 0.
*
*  LDC     (input) INTEGER
*          The leading dimension of the array C.
*          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
*
*  WORK    (workspace) COMPLEX array, dimension (max(M,N))
*
*  RWORK   (workspace) REAL array, dimension (max(M,N))
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*

*  =====================================================================
*

```
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## cgbcon

```USAGE:
rcond, info = NumRu::Lapack.cgbcon( norm, kl, ku, ab, ipiv, anorm, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, RWORK, INFO )

*  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)) ).
*

*  Arguments
*  =========
*
*  NORM    (input) 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       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  AB      (input) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          The pivot indices; for 1 <= i <= N, row i of the matrix was
*          interchanged with row IPIV(i).
*
*  ANORM   (input) 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   (output) REAL
*          The reciprocal of the condition number of the matrix A,
*          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
*  WORK    (workspace) COMPLEX array, dimension (2*N)
*
*  RWORK   (workspace) REAL array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*

*  =====================================================================
*

```
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## cgbequ

```USAGE:
r, c, rowcnd, colcnd, amax, info = NumRu::Lapack.cgbequ( m, kl, ku, ab, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO )

*  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.
*

*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  AB      (input) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*
*  R       (output) REAL array, dimension (M)
*          If INFO = 0, or INFO > M, R contains the row scale factors
*          for A.
*
*  C       (output) REAL array, dimension (N)
*          If INFO = 0, C contains the column scale factors for A.
*
*  ROWCND  (output) 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  (output) 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    (output) 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    (output) 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
*

*  =====================================================================
*

```
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## cgbequb

```USAGE:
r, c, rowcnd, colcnd, amax, info = NumRu::Lapack.cgbequb( kl, ku, ab, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO )

*  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
*
*  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.  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
*

*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  AB      (input) DOUBLE PRECISION 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    (input) INTEGER
*          The leading dimension of the array A.  LDAB >= max(1,M).
*
*  R       (output) REAL array, dimension (M)
*          If INFO = 0 or INFO > M, R contains the row scale factors
*          for A.
*
*  C       (output) REAL array, dimension (N)
*          If INFO = 0,  C contains the column scale factors for A.
*
*  ROWCND  (output) 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  (output) 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    (output) 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    (output) 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
*

*  =====================================================================
*

```
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## cgbrfs

```USAGE:
ferr, berr, info, x = NumRu::Lapack.cgbrfs( trans, kl, ku, ab, afb, ipiv, b, x, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )

*  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.
*

*  Arguments
*  =========
*
*  TRANS   (input) 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       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  AB      (input) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*
*  AFB     (input) 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   (input) INTEGER
*          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
*
*  IPIV    (input) 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       (input) COMPLEX array, dimension (LDB,NRHS)
*          The right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (input/output) COMPLEX array, dimension (LDX,NRHS)
*          On entry, the solution matrix X, as computed by CGBTRS.
*          On exit, the improved solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  FERR    (output) 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    (output) 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    (workspace) COMPLEX array, dimension (2*N)
*
*  RWORK   (workspace) REAL array, dimension (N)
*
*  INFO    (output) 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.
*

*  =====================================================================
*

```
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## cgbrfsx

```USAGE:
rcond, berr, err_bnds_norm, err_bnds_comp, info, r, c, x, params = NumRu::Lapack.cgbrfsx( trans, equed, kl, ku, ab, afb, ipiv, r, c, b, x, params, [:usage => usage, :help => help])

FORTRAN MANUAL
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 )

*     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.
*

*     Arguments
*     =========
*
*     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.
*
*     TRANS   (input) CHARACTER*1
*     Specifies the form of the system of equations:
*       = 'N':  A * X = B     (No transpose)
*       = 'T':  A**T * X = B  (Transpose)
*       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*
*     EQUED   (input) 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       (input) INTEGER
*     The order of the matrix A.  N >= 0.
*
*     KL      (input) INTEGER
*     The number of subdiagonals within the band of A.  KL >= 0.
*
*     KU      (input) INTEGER
*     The number of superdiagonals within the band of A.  KU >= 0.
*
*     NRHS    (input) INTEGER
*     The number of right hand sides, i.e., the number of columns
*     of the matrices B and X.  NRHS >= 0.
*
*     AB      (input) DOUBLE PRECISION 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    (input) INTEGER
*     The leading dimension of the array AB.  LDAB >= KL+KU+1.
*
*     AFB     (input) 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.
*
*     LDAFB   (input) INTEGER
*     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
*
*     IPIV    (input) INTEGER array, dimension (N)
*     The pivot indices from SGETRF; for 1<=i<=N, row i of the
*     matrix was interchanged with row IPIV(i).
*
*     R       (input or output) 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       (input or output) 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       (input) REAL array, dimension (LDB,NRHS)
*     The right hand side matrix B.
*
*     LDB     (input) INTEGER
*     The leading dimension of the array B.  LDB >= max(1,N).
*
*     X       (input/output) REAL array, dimension (LDX,NRHS)
*     On entry, the solution matrix X, as computed by SGETRS.
*     On exit, the improved solution matrix X.
*
*     LDX     (input) INTEGER
*     The leading dimension of the array X.  LDX >= max(1,N).
*
*     RCOND   (output) 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    (output) 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 (input) 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  (output) 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  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
*     For each right-hand side, this array contains information about
*     various error bounds and condition numbers corresponding to the
*     componentwise relative error, which is defined as follows:
*
*     Componentwise relative error in the ith solution vector:
*                    abs(XTRUE(j,i) - X(j,i))
*             max_j ----------------------
*                         abs(X(j,i))
*
*     The array is indexed by the right-hand side i (on which the
*     componentwise relative error depends), and the type of error
*     information as described below. There currently are up to three
*     pieces of information returned for each right-hand side. If
*     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
*     the first (:,N_ERR_BNDS) entries are returned.
*
*     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
*     right-hand side.
*
*     The second index in ERR_BNDS_COMP(:,err) contains the following
*     three fields:
*     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*              reciprocal condition number is less than the threshold
*              sqrt(n) * slamch('Epsilon').
*
*     err = 2 "Guaranteed" error bound: The estimated forward error,
*              almost certainly within a factor of 10 of the true error
*              so long as the next entry is greater than the threshold
*              sqrt(n) * slamch('Epsilon'). This error bound should only
*              be trusted if the previous boolean is true.
*
*     err = 3  Reciprocal condition number: Estimated componentwise
*              reciprocal condition number.  Compared with the threshold
*              sqrt(n) * slamch('Epsilon') to determine if the error
*              estimate is "guaranteed". These reciprocal condition
*              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*              appropriately scaled matrix Z.
*              Let Z = S*(A*diag(x)), where x is the solution for the
*              current right-hand side and S scales each row of
*              A*diag(x) by a power of the radix so all absolute row
*              sums of Z are approximately 1.
*
*     See Lapack Working Note 165 for further details and extra
*     cautions.
*
*     NPARAMS (input) INTEGER
*     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
*     PARAMS array is never referenced and default values are used.
*
*     PARAMS  (input / output) REAL array, dimension NPARAMS
*     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
*     that entry will be filled with default value used for that
*     parameter.  Only positions up to NPARAMS are accessed; defaults
*     are used for higher-numbered parameters.
*
*       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*            refinement or not.
*         Default: 1.0
*            = 0.0 : No refinement is performed, and no error bounds are
*                    computed.
*            = 1.0 : Use the double-precision refinement algorithm,
*                    possibly with doubled-single computations if the
*                    compilation environment does not support DOUBLE
*                    PRECISION.
*              (other values are reserved for future use)
*
*       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
*            computations allowed for refinement.
*         Default: 10
*         Aggressive: Set to 100 to permit convergence using approximate
*                     factorizations or factorizations other than LU. If
*                     the factorization uses a technique other than
*                     Gaussian elimination, the guarantees in
*                     err_bnds_norm and err_bnds_comp may no longer be
*                     trustworthy.
*
*       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
*            will attempt to find a solution with small componentwise
*            relative error in the double-precision algorithm.  Positive
*            is true, 0.0 is false.
*         Default: 1.0 (attempt componentwise convergence)
*
*     WORK    (workspace) COMPLEX array, dimension (2*N)
*
*     RWORK   (workspace) REAL array, dimension (2*N)
*
*     INFO    (output) 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.
*

*     ==================================================================
*

```
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## cgbsv

```USAGE:
ipiv, info, ab, b = NumRu::Lapack.cgbsv( kl, ku, ab, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

*  Purpose
*  =======
*
*  CGBSV computes the solution to a complex system of linear equations
*  A * X = B, where A is a band matrix of order N with KL subdiagonals
*  and KU superdiagonals, and X and B are N-by-NRHS matrices.
*
*  The LU decomposition with partial pivoting and row interchanges is
*  used to factor A as A = L * U, where L is a product of permutation
*  and unit lower triangular matrices with KL subdiagonals, and U is
*  upper triangular with KL+KU superdiagonals.  The factored form of A
*  is then used to solve the system of equations A * X = B.
*

*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  AB      (input/output) 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(N,j+KL)
*          On exit, details of the factorization: U is stored as an
*          upper triangular band matrix with KL+KU superdiagonals in
*          rows 1 to KL+KU+1, and the multipliers used during the
*          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
*          See below for further details.
*
*  LDAB    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*
*  IPIV    (output) INTEGER array, dimension (N)
*          The pivot indices that define the permutation matrix P;
*          row i of the matrix was interchanged with row IPIV(i).
*
*  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
*          On entry, the N-by-NRHS right hand side matrix B.
*          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
*                has been completed, but the factor U is exactly
*                singular, and the solution has not been computed.
*

*  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.
*
*  =====================================================================
*
*     .. External Subroutines ..
EXTERNAL           CGBTRF, CGBTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## cgbsvx

```USAGE:
x, rcond, ferr, berr, rwork, info, ab, afb, ipiv, equed, r, c, b = NumRu::Lapack.cgbsvx( fact, trans, kl, ku, ab, b, [:afb => afb, :ipiv => ipiv, :equed => equed, :r => r, :c => c, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

*  Purpose
*  =======
*
*  CGBSVX uses the LU factorization to compute the solution to a complex
*  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
*  where A is a band matrix of order N with KL subdiagonals and KU
*  superdiagonals, and X and B are N-by-NRHS matrices.
*
*  Error bounds on the solution and a condition estimate are also
*  provided.
*
*  Description
*  ===========
*
*  The following steps are performed by this subroutine:
*
*  1. If FACT = 'E', real scaling factors are computed to equilibrate
*     the system:
*        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
*        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*     Whether or not the system will be equilibrated depends on the
*     scaling of the matrix A, but if equilibration is used, A is
*     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*     or diag(C)*B (if TRANS = 'T' or 'C').
*
*  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
*     matrix A (after equilibration if FACT = 'E') as
*        A = L * U,
*     where L is a product of permutation and unit lower triangular
*     matrices with KL subdiagonals, and U is upper triangular with
*     KL+KU superdiagonals.
*
*  3. If some U(i,i)=0, so that U is exactly singular, then the routine
*     returns with INFO = i. Otherwise, the factored form of A is used
*     to estimate the condition number of the matrix A.  If the
*     reciprocal of the condition number is less than machine precision,
*     INFO = N+1 is returned as a warning, but the routine still goes on
*     to solve for X and compute error bounds as described below.
*
*  4. The system of equations is solved for X using the factored form
*     of A.
*
*  5. Iterative refinement is applied to improve the computed solution
*     matrix and calculate error bounds and backward error estimates
*     for it.
*
*  6. If equilibration was used, the matrix X is premultiplied by
*     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*     that it solves the original system before equilibration.
*

*  Arguments
*  =========
*
*  FACT    (input) CHARACTER*1
*          Specifies whether or not the factored form of the matrix A is
*          supplied on entry, and if not, whether the matrix A should be
*          equilibrated before it is factored.
*          = 'F':  On entry, AFB and IPIV contain the factored form of
*                  A.  If EQUED is not 'N', the matrix A has been
*                  equilibrated with scaling factors given by R and C.
*                  AB, AFB, and IPIV are not modified.
*          = 'N':  The matrix A will be copied to AFB and factored.
*          = 'E':  The matrix A will be equilibrated if necessary, then
*                  copied to AFB and factored.
*
*  TRANS   (input) 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       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  AB      (input/output) 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)
*
*          If FACT = 'F' and EQUED is not 'N', then A must have been
*          equilibrated by the scaling factors in R and/or C.  AB is not
*          modified if FACT = 'F' or 'N', or if FACT = 'E' and
*          EQUED = 'N' on exit.
*
*          On exit, if EQUED .ne. 'N', A is scaled as follows:
*          EQUED = 'R':  A := diag(R) * A
*          EQUED = 'C':  A := A * diag(C)
*          EQUED = 'B':  A := diag(R) * A * diag(C).
*
*  LDAB    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KL+KU+1.
*
*  AFB     (input or output) COMPLEX array, dimension (LDAFB,N)
*          If FACT = 'F', then AFB is an input argument and on entry
*          contains details of the LU factorization of the band matrix
*          A, as computed by 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.  If EQUED .ne. 'N', then AFB is
*          the factored form of the equilibrated matrix A.
*
*          If FACT = 'N', then AFB is an output argument and on exit
*          returns details of the LU factorization of A.
*
*          If FACT = 'E', then AFB is an output argument and on exit
*          returns details of the LU factorization of the equilibrated
*          matrix A (see the description of AB for the form of the
*          equilibrated matrix).
*
*  LDAFB   (input) INTEGER
*          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
*
*  IPIV    (input or output) INTEGER array, dimension (N)
*          If FACT = 'F', then IPIV is an input argument and on entry
*          contains the pivot indices from the factorization A = L*U
*          as computed by CGBTRF; row i of the matrix was interchanged
*          with row IPIV(i).
*
*          If FACT = 'N', then IPIV is an output argument and on exit
*          contains the pivot indices from the factorization A = L*U
*          of the original matrix A.
*
*          If FACT = 'E', then IPIV is an output argument and on exit
*          contains the pivot indices from the factorization A = L*U
*          of the equilibrated matrix A.
*
*  EQUED   (input or output) CHARACTER*1
*          Specifies the form of equilibration that was done.
*          = 'N':  No equilibration (always true if FACT = 'N').
*          = 'R':  Row equilibration, i.e., A has been premultiplied by
*                  diag(R).
*          = 'C':  Column equilibration, i.e., A has been postmultiplied
*                  by diag(C).
*          = 'B':  Both row and column equilibration, i.e., A has been
*                  replaced by diag(R) * A * diag(C).
*          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*          output argument.
*
*  R       (input or output) REAL array, dimension (N)
*          The row scale factors for A.  If EQUED = 'R' or 'B', A is
*          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*          is not accessed.  R is an input argument if FACT = 'F';
*          otherwise, R is an output argument.  If FACT = 'F' and
*          EQUED = 'R' or 'B', each element of R must be positive.
*
*  C       (input or output) REAL array, dimension (N)
*          The column scale factors for A.  If EQUED = 'C' or 'B', A is
*          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*          is not accessed.  C is an input argument if FACT = 'F';
*          otherwise, C is an output argument.  If FACT = 'F' and
*          EQUED = 'C' or 'B', each element of C must be positive.
*
*  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
*          On entry, the right hand side matrix B.
*          On exit,
*          if EQUED = 'N', B is not modified;
*          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*          diag(R)*B;
*          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*          overwritten by diag(C)*B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (output) COMPLEX array, dimension (LDX,NRHS)
*          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
*          to the original system of equations.  Note that A and B are
*          modified on exit if EQUED .ne. 'N', and the solution to the
*          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
*          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
*          and EQUED = 'R' or 'B'.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  RCOND   (output) REAL
*          The estimate of the reciprocal condition number of the matrix
*          A after equilibration (if done).  If RCOND is less than the
*          machine precision (in particular, if RCOND = 0), the matrix
*          is singular to working precision.  This condition is
*          indicated by a return code of INFO > 0.
*
*  FERR    (output) 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    (output) 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    (workspace) COMPLEX array, dimension (2*N)
*
*  RWORK   (workspace/output) REAL array, dimension (N)
*          On exit, RWORK(1) contains the reciprocal pivot growth
*          factor norm(A)/norm(U). The "max absolute element" norm is
*          used. If RWORK(1) is much less than 1, then the stability
*          of the LU factorization of the (equilibrated) matrix A
*          could be poor. This also means that the solution X, condition
*          estimator RCOND, and forward error bound FERR could be
*          unreliable. If factorization fails with 0 0:  if INFO = i, and i is
*                <= N:  U(i,i) is exactly zero.  The factorization
*                       has been completed, but the factor U is exactly
*                       singular, so the solution and error bounds
*                       could not be computed. RCOND = 0 is returned.
*                = N+1: U is nonsingular, but RCOND is less than machine
*                       precision, meaning that the matrix is singular
*                       to working precision.  Nevertheless, the
*                       solution and error bounds are computed because
*                       there are a number of situations where the
*                       computed solution can be more accurate than the
*                       value of RCOND would suggest.
*

*  =====================================================================
*  Moved setting of INFO = N+1 so INFO does not subsequently get
*  overwritten.  Sven, 17 Mar 05.
*  =====================================================================
*

```
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## cgbsvxx

```USAGE:
x, rcond, rpvgrw, berr, err_bnds_norm, err_bnds_comp, info, ab, afb, ipiv, equed, r, c, b, params = NumRu::Lapack.cgbsvxx( fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b, params, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )

*     Purpose
*     =======
*
*     CGBSVXX uses the LU factorization to compute the solution to a
*     complex system of linear equations  A * X = B,  where A is an
*     N-by-N matrix and X and B are N-by-NRHS matrices.
*
*     If requested, both normwise and maximum componentwise error bounds
*     are returned. CGBSVXX will return a solution with a tiny
*     guaranteed error (O(eps) where eps is the working machine
*     precision) unless the matrix is very ill-conditioned, in which
*     case a warning is returned. Relevant condition numbers also are
*     calculated and returned.
*
*     CGBSVXX accepts user-provided factorizations and equilibration
*     factors; see the definitions of the FACT and EQUED options.
*     Solving with refinement and using a factorization from a previous
*     CGBSVXX call will also produce a solution with either O(eps)
*     errors or warnings, but we cannot make that claim for general
*     user-provided factorizations and equilibration factors if they
*     differ from what CGBSVXX would itself produce.
*
*     Description
*     ===========
*
*     The following steps are performed:
*
*     1. If FACT = 'E', real scaling factors are computed to equilibrate
*     the system:
*
*       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
*       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*
*     Whether or not the system will be equilibrated depends on the
*     scaling of the matrix A, but if equilibration is used, A is
*     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*     or diag(C)*B (if TRANS = 'T' or 'C').
*
*     2. If FACT = 'N' or 'E', the LU decomposition is used to factor
*     the matrix A (after equilibration if FACT = 'E') as
*
*       A = P * L * U,
*
*     where P is a permutation matrix, L is a unit lower triangular
*     matrix, and U is upper triangular.
*
*     3. If some U(i,i)=0, so that U is exactly singular, then the
*     routine returns with INFO = i. Otherwise, the factored form of A
*     is used to estimate the condition number of the matrix A (see
*     argument RCOND). If the reciprocal of the condition number is less
*     than machine precision, the routine still goes on to solve for X
*     and compute error bounds as described below.
*
*     4. The system of equations is solved for X using the factored form
*     of A.
*
*     5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
*     the routine will use iterative refinement to try to get a small
*     error and error bounds.  Refinement calculates the residual to at
*     least twice the working precision.
*
*     6. If equilibration was used, the matrix X is premultiplied by
*     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*     that it solves the original system before equilibration.
*

*     Arguments
*     =========
*
*     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.
*
*     FACT    (input) CHARACTER*1
*     Specifies whether or not the factored form of the matrix A is
*     supplied on entry, and if not, whether the matrix A should be
*     equilibrated before it is factored.
*       = 'F':  On entry, AF and IPIV contain the factored form of A.
*               If EQUED is not 'N', the matrix A has been
*               equilibrated with scaling factors given by R and C.
*               A, AF, and IPIV are not modified.
*       = 'N':  The matrix A will be copied to AF and factored.
*       = 'E':  The matrix A will be equilibrated if necessary, then
*               copied to AF and factored.
*
*     TRANS   (input) 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       (input) INTEGER
*     The number of linear equations, i.e., the order of the
*     matrix A.  N >= 0.
*
*     KL      (input) INTEGER
*     The number of subdiagonals within the band of A.  KL >= 0.
*
*     KU      (input) INTEGER
*     The number of superdiagonals within the band of A.  KU >= 0.
*
*     NRHS    (input) INTEGER
*     The number of right hand sides, i.e., the number of columns
*     of the matrices B and X.  NRHS >= 0.
*
*     AB      (input/output) REAL array, dimension (LDAB,N)
*     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*     The j-th column of A is stored in the j-th column of the
*     array AB as follows:
*     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*
*     If FACT = 'F' and EQUED is not 'N', then AB must have been
*     equilibrated by the scaling factors in R and/or C.  AB is not
*     modified if FACT = 'F' or 'N', or if FACT = 'E' and
*     EQUED = 'N' on exit.
*
*     On exit, if EQUED .ne. 'N', A is scaled as follows:
*     EQUED = 'R':  A := diag(R) * A
*     EQUED = 'C':  A := A * diag(C)
*     EQUED = 'B':  A := diag(R) * A * diag(C).
*
*     LDAB    (input) INTEGER
*     The leading dimension of the array AB.  LDAB >= KL+KU+1.
*
*     AFB     (input or output) REAL array, dimension (LDAFB,N)
*     If FACT = 'F', then AFB is an input argument and on entry
*     contains details of the LU factorization of the band matrix
*     A, as computed by 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.  If EQUED .ne. 'N', then AFB is
*     the factored form of the equilibrated matrix A.
*
*     If FACT = 'N', then AF is an output argument and on exit
*     returns the factors L and U from the factorization A = P*L*U
*     of the original matrix A.
*
*     If FACT = 'E', then AF is an output argument and on exit
*     returns the factors L and U from the factorization A = P*L*U
*     of the equilibrated matrix A (see the description of A for
*     the form of the equilibrated matrix).
*
*     LDAFB   (input) INTEGER
*     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
*
*     IPIV    (input or output) INTEGER array, dimension (N)
*     If FACT = 'F', then IPIV is an input argument and on entry
*     contains the pivot indices from the factorization A = P*L*U
*     as computed by SGETRF; row i of the matrix was interchanged
*     with row IPIV(i).
*
*     If FACT = 'N', then IPIV is an output argument and on exit
*     contains the pivot indices from the factorization A = P*L*U
*     of the original matrix A.
*
*     If FACT = 'E', then IPIV is an output argument and on exit
*     contains the pivot indices from the factorization A = P*L*U
*     of the equilibrated matrix A.
*
*     EQUED   (input or output) CHARACTER*1
*     Specifies the form of equilibration that was done.
*       = 'N':  No equilibration (always true if FACT = 'N').
*       = 'R':  Row equilibration, i.e., A has been premultiplied by
*               diag(R).
*       = 'C':  Column equilibration, i.e., A has been postmultiplied
*               by diag(C).
*       = 'B':  Both row and column equilibration, i.e., A has been
*               replaced by diag(R) * A * diag(C).
*     EQUED is an input argument if FACT = 'F'; otherwise, it is an
*     output argument.
*
*     R       (input or output) 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       (input or output) 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       (input/output) REAL array, dimension (LDB,NRHS)
*     On entry, the N-by-NRHS right hand side matrix B.
*     On exit,
*     if EQUED = 'N', B is not modified;
*     if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*        diag(R)*B;
*     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*        overwritten by diag(C)*B.
*
*     LDB     (input) INTEGER
*     The leading dimension of the array B.  LDB >= max(1,N).
*
*     X       (output) REAL array, dimension (LDX,NRHS)
*     If INFO = 0, the N-by-NRHS solution matrix X to the original
*     system of equations.  Note that A and B are modified on exit
*     if EQUED .ne. 'N', and the solution to the equilibrated system is
*     inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
*     inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
*
*     LDX     (input) INTEGER
*     The leading dimension of the array X.  LDX >= max(1,N).
*
*     RCOND   (output) REAL
*     Reciprocal scaled condition number.  This is an estimate of the
*     reciprocal Skeel condition number of the matrix A after
*     equilibration (if done).  If this is less than the machine
*     precision (in particular, if it is zero), the matrix is singular
*     to working precision.  Note that the error may still be small even
*     if this number is very small and the matrix appears ill-
*     conditioned.
*
*     RPVGRW  (output) REAL
*     Reciprocal pivot growth.  On exit, this contains the reciprocal
*     pivot growth factor norm(A)/norm(U). The "max absolute element"
*     norm is used.  If this is much less than 1, then the stability of
*     the LU factorization of the (equilibrated) matrix A could be poor.
*     This also means that the solution X, estimated condition numbers,
*     and error bounds could be unreliable. If factorization fails with
*     0 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.
*

*     ==================================================================
*

```
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## cgbtf2

```USAGE:
ipiv, info, ab = NumRu::Lapack.cgbtf2( m, kl, ku, ab, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )

*  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.
*

*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  AB      (input/output) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*
*  IPIV    (output) 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    (output) 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.
*

*  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.
*
*  =====================================================================
*

```
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## cgbtrf

```USAGE:
ipiv, info, ab = NumRu::Lapack.cgbtrf( m, kl, ku, ab, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )

*  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.
*

*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  AB      (input/output) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*
*  IPIV    (output) 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    (output) 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.
*

*  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.
*
*  =====================================================================
*

```
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## cgbtrs

```USAGE:
info, b = NumRu::Lapack.cgbtrs( trans, kl, ku, ab, ipiv, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

*  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.
*

*  Arguments
*  =========
*
*  TRANS   (input) 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       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  KL      (input) INTEGER
*          The number of subdiagonals within the band of A.  KL >= 0.
*
*  KU      (input) INTEGER
*          The number of superdiagonals within the band of A.  KU >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  AB      (input) 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    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          The pivot indices; for 1 <= i <= N, row i of the matrix was
*          interchanged with row IPIV(i).
*
*  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
*          On entry, the right hand side matrix B.
*          On exit, the solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

*  =====================================================================
*

```
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