# COMPLEX routines for symmetric or Hermitian positive definite matrix

## cpocon

```USAGE:
rcond, info = NumRu::Lapack.cpocon( uplo, a, anorm, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO )

*  Purpose
*  =======
*
*  CPOCON estimates the reciprocal of the condition number (in the
*  1-norm) of a complex Hermitian positive definite matrix using the
*  Cholesky factorization A = U**H*U or A = L*L**H computed by CPOTRF.
*
*  An estimate is obtained for norm(inv(A)), and the reciprocal of the
*  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**H*U or A = L*L**H, as computed by CPOTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  ANORM   (input) REAL
*          The 1-norm (or infinity-norm) of the Hermitian matrix A.
*
*  RCOND   (output) REAL
*          The reciprocal of the condition number of the matrix A,
*          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*          estimate of the 1-norm of inv(A) computed in this routine.
*
*  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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## cpoequ

```USAGE:
s, scond, amax, info = NumRu::Lapack.cpoequ( a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )

*  Purpose
*  =======
*
*  CPOEQU computes row and column scalings intended to equilibrate a
*  Hermitian positive definite matrix A and reduce its condition number
*  (with respect to the two-norm).  S contains the scale factors,
*  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*  choice of S puts the condition number of B within a factor N of the
*  smallest possible condition number over all possible diagonal
*  scalings.
*

*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The N-by-N Hermitian positive definite matrix whose scaling
*          factors are to be computed.  Only the diagonal elements of A
*          are referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  S       (output) REAL array, dimension (N)
*          If INFO = 0, S contains the scale factors for A.
*
*  SCOND   (output) REAL
*          If INFO = 0, S contains the ratio of the smallest S(i) to
*          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*          large nor too small, it is not worth scaling by S.
*
*  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, the i-th diagonal element is nonpositive.
*

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

```
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## cpoequb

```USAGE:
s, scond, amax, info = NumRu::Lapack.cpoequb( a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )

*  Purpose
*  =======
*
*  CPOEQUB computes row and column scalings intended to equilibrate a
*  symmetric positive definite matrix A and reduce its condition number
*  (with respect to the two-norm).  S contains the scale factors,
*  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
*  choice of S puts the condition number of B within a factor N of the
*  smallest possible condition number over all possible diagonal
*  scalings.
*

*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The N-by-N symmetric positive definite matrix whose scaling
*          factors are to be computed.  Only the diagonal elements of A
*          are referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  S       (output) REAL array, dimension (N)
*          If INFO = 0, S contains the scale factors for A.
*
*  SCOND   (output) REAL
*          If INFO = 0, S contains the ratio of the smallest S(i) to
*          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
*          large nor too small, it is not worth scaling by S.
*
*  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, the i-th diagonal element is nonpositive.
*

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

```
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## cporfs

```USAGE:
ferr, berr, info, x = NumRu::Lapack.cporfs( uplo, a, af, b, x, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )

*  Purpose
*  =======
*
*  CPORFS improves the computed solution to a system of linear
*  equations when the coefficient matrix is Hermitian positive definite,
*  and provides error bounds and backward error estimates for the
*  solution.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
*          upper triangular part of A contains the upper triangular part
*          of the matrix A, and the strictly lower triangular part of A
*          is not referenced.  If UPLO = 'L', the leading N-by-N lower
*          triangular part of A contains the lower triangular part of
*          the matrix A, and the strictly upper triangular part of A is
*          not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  AF      (input) COMPLEX array, dimension (LDAF,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**H*U or A = L*L**H, as computed by CPOTRF.
*
*  LDAF    (input) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  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 CPOTRS.
*          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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## cporfsx

```USAGE:
rcond, berr, err_bnds_norm, err_bnds_comp, info, s, x, params = NumRu::Lapack.cporfsx( uplo, equed, a, af, s, b, x, params, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )

*     Purpose
*     =======
*
*     CPORFSX improves the computed solution to a system of linear
*     equations when the coefficient matrix is symmetric positive
*     definite, 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 and S
*     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.
*
*     UPLO    (input) CHARACTER*1
*       = 'U':  Upper triangle of A is stored;
*       = 'L':  Lower triangle of A is stored.
*
*     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
*       = 'Y':  Both row and column equilibration, i.e., A has been
*               replaced by diag(S) * A * diag(S).
*               The right hand side B has been changed accordingly.
*
*     N       (input) INTEGER
*     The order of the matrix A.  N >= 0.
*
*     NRHS    (input) INTEGER
*     The number of right hand sides, i.e., the number of columns
*     of the matrices B and X.  NRHS >= 0.
*
*     A       (input) COMPLEX array, dimension (LDA,N)
*     The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
*     upper triangular part of A contains the upper triangular part
*     of the matrix A, and the strictly lower triangular part of A
*     is not referenced.  If UPLO = 'L', the leading N-by-N lower
*     triangular part of A contains the lower triangular part of
*     the matrix A, and the strictly upper triangular part of A is
*     not referenced.
*
*     LDA     (input) INTEGER
*     The leading dimension of the array A.  LDA >= max(1,N).
*
*     AF      (input) COMPLEX array, dimension (LDAF,N)
*     The triangular factor U or L from the Cholesky factorization
*     A = U**T*U or A = L*L**T, as computed by SPOTRF.
*
*     LDAF    (input) INTEGER
*     The leading dimension of the array AF.  LDAF >= max(1,N).
*
*     S       (input or output) REAL array, dimension (N)
*     The row scale factors for A.  If EQUED = 'Y', A is multiplied on
*     the left and right by diag(S).  S is an input argument if FACT =
*     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
*     = 'Y', each element of S must be positive.  If S is output, each
*     element of S is a power of the radix. If S is input, each element
*     of S 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) 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 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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## cposv

```USAGE:
info, a, b = NumRu::Lapack.cposv( uplo, a, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )

*  Purpose
*  =======
*
*  CPOSV computes the solution to a complex system of linear equations
*     A * X = B,
*  where A is an N-by-N Hermitian positive definite matrix and X and B
*  are N-by-NRHS matrices.
*
*  The Cholesky decomposition is used to factor A as
*     A = U**H* U,  if UPLO = 'U', or
*     A = L * L**H,  if UPLO = 'L',
*  where U is an upper triangular matrix and  L is a lower triangular
*  matrix.  The factored form of A is then used to solve the system of
*  equations A * X = B.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  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, the leading minor of order i of A is not
*                positive definite, so the factorization could not be
*                completed, and the solution has not been computed.
*

*  =====================================================================
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           CPOTRF, CPOTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## cposvx

```USAGE:
x, rcond, ferr, berr, info, a, af, equed, s, b = NumRu::Lapack.cposvx( fact, uplo, a, af, equed, s, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

*  Purpose
*  =======
*
*  CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
*  compute the solution to a complex system of linear equations
*     A * X = B,
*  where A is an N-by-N Hermitian positive definite matrix 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:
*
*  1. If FACT = 'E', real scaling factors are computed to equilibrate
*     the system:
*        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B.
*
*  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*     factor the matrix A (after equilibration if FACT = 'E') as
*        A = U**H* U,  if UPLO = 'U', or
*        A = L * L**H,  if UPLO = 'L',
*     where U is an upper triangular matrix and L is a lower triangular
*     matrix.
*
*  3. If the leading i-by-i principal minor is not positive definite,
*     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(S) 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, AF contains the factored form of A.
*                  If EQUED = 'Y', the matrix A has been equilibrated
*                  with scaling factors given by S.  A and AF will not
*                  be 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.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the Hermitian matrix A, except if FACT = 'F' and
*          EQUED = 'Y', then A must contain the equilibrated matrix
*          diag(S)*A*diag(S).  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.  A is not modified if
*          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*
*          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*          diag(S)*A*diag(S).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  AF      (input or output) COMPLEX array, dimension (LDAF,N)
*          If FACT = 'F', then AF is an input argument and on entry
*          contains the triangular factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H, in the same storage
*          format as A.  If EQUED .ne. 'N', then AF is the factored form
*          of the equilibrated matrix diag(S)*A*diag(S).
*
*          If FACT = 'N', then AF is an output argument and on exit
*          returns the triangular factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H of the original
*          matrix A.
*
*          If FACT = 'E', then AF is an output argument and on exit
*          returns the triangular factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H of the equilibrated
*          matrix A (see the description of A for the form of the
*          equilibrated matrix).
*
*  LDAF    (input) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  EQUED   (input or output) CHARACTER*1
*          Specifies the form of equilibration that was done.
*          = 'N':  No equilibration (always true if FACT = 'N').
*          = 'Y':  Equilibration was done, i.e., A has been replaced by
*                  diag(S) * A * diag(S).
*          EQUED is an input argument if FACT = 'F'; otherwise, it is an
*          output argument.
*
*  S       (input or output) REAL array, dimension (N)
*          The scale factors for A; not accessed if EQUED = 'N'.  S is
*          an input argument if FACT = 'F'; otherwise, S is an output
*          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
*          must be positive.
*
*  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
*          On entry, the N-by-NRHS righthand side matrix B.
*          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
*          B is overwritten by diag(S) * 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 if EQUED = 'Y',
*          A and B are modified on exit, and the solution to the
*          equilibrated system is inv(diag(S))*X.
*
*  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) REAL array, dimension (N)
*
*  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
*                <= N:  the leading minor of order i of A is
*                       not positive definite, so the factorization
*                       could not be completed, and the solution has not
*                       been 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.
*

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

```
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## cposvxx

```USAGE:
x, rcond, rpvgrw, berr, err_bnds_norm, err_bnds_comp, info, a, af, equed, s, b, params = NumRu::Lapack.cposvxx( fact, uplo, a, af, equed, s, b, params, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )

*     Purpose
*     =======
*
*     CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
*     to compute the solution to a complex system of linear equations
*     A * X = B, where A is an N-by-N symmetric positive definite matrix
*     and X and B are N-by-NRHS matrices.
*
*     If requested, both normwise and maximum componentwise error bounds
*     are returned. CPOSVXX 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.
*
*     CPOSVXX 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
*     CPOSVXX 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 CPOSVXX would itself produce.
*
*     Description
*     ===========
*
*     The following steps are performed:
*
*     1. If FACT = 'E', real scaling factors are computed to equilibrate
*     the system:
*
*       diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.
*
*     2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
*     factor the matrix A (after equilibration if FACT = 'E') as
*        A = U**T* U,  if UPLO = 'U', or
*        A = L * L**T,  if UPLO = 'L',
*     where U is an upper triangular matrix and L is a lower triangular
*     matrix.
*
*     3. If the leading i-by-i principal minor is not positive definite,
*     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(S) 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 contains the factored form of A.
*               If EQUED is not 'N', the matrix A has been
*               equilibrated with scaling factors given by S.
*               A and AF 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.
*
*     UPLO    (input) CHARACTER*1
*       = 'U':  Upper triangle of A is stored;
*       = 'L':  Lower triangle of A is stored.
*
*     N       (input) INTEGER
*     The number of linear equations, i.e., the order of the
*     matrix A.  N >= 0.
*
*     NRHS    (input) INTEGER
*     The number of right hand sides, i.e., the number of columns
*     of the matrices B and X.  NRHS >= 0.
*
*     A       (input/output) COMPLEX array, dimension (LDA,N)
*     On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
*     'Y', then A must contain the equilibrated matrix
*     diag(S)*A*diag(S).  If UPLO = 'U', the leading N-by-N upper
*     triangular part of A contains the upper triangular part of the
*     matrix A, and the strictly lower triangular part of A is not
*     referenced.  If UPLO = 'L', the leading N-by-N lower triangular
*     part of A contains the lower triangular part of the matrix A, and
*     the strictly upper triangular part of A is not referenced.  A is
*     not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
*     'N' on exit.
*
*     On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
*     diag(S)*A*diag(S).
*
*     LDA     (input) INTEGER
*     The leading dimension of the array A.  LDA >= max(1,N).
*
*     AF      (input or output) COMPLEX array, dimension (LDAF,N)
*     If FACT = 'F', then AF is an input argument and on entry
*     contains the triangular factor U or L from the Cholesky
*     factorization A = U**T*U or A = L*L**T, in the same storage
*     format as A.  If EQUED .ne. 'N', then AF is the factored
*     form of the equilibrated matrix diag(S)*A*diag(S).
*
*     If FACT = 'N', then AF is an output argument and on exit
*     returns the triangular factor U or L from the Cholesky
*     factorization A = U**T*U or A = L*L**T of the original
*     matrix A.
*
*     If FACT = 'E', then AF is an output argument and on exit
*     returns the triangular factor U or L from the Cholesky
*     factorization A = U**T*U or A = L*L**T of the equilibrated
*     matrix A (see the description of A for the form of the
*     equilibrated matrix).
*
*     LDAF    (input) INTEGER
*     The leading dimension of the array AF.  LDAF >= max(1,N).
*
*     EQUED   (input or output) CHARACTER*1
*     Specifies the form of equilibration that was done.
*       = 'N':  No equilibration (always true if FACT = 'N').
*       = 'Y':  Both row and column equilibration, i.e., A has been
*               replaced by diag(S) * A * diag(S).
*     EQUED is an input argument if FACT = 'F'; otherwise, it is an
*     output argument.
*
*     S       (input or output) REAL array, dimension (N)
*     The row scale factors for A.  If EQUED = 'Y', A is multiplied on
*     the left and right by diag(S).  S is an input argument if FACT =
*     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
*     = 'Y', each element of S must be positive.  If S is output, each
*     element of S is a power of the radix. If S is input, each element
*     of S 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) COMPLEX 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 EQUED = 'Y', B is overwritten by diag(S)*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, 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(S))*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.
*
*     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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## cpotf2

```USAGE:
info, a = NumRu::Lapack.cpotf2( uplo, a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOTF2( UPLO, N, A, LDA, INFO )

*  Purpose
*  =======
*
*  CPOTF2 computes the Cholesky factorization of a complex Hermitian
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U' * U ,  if UPLO = 'U', or
*     A = L  * L',  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the unblocked version of the algorithm, calling Level 2 BLAS.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          Hermitian matrix A is stored.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
*          n by n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n by n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U'*U  or A = L*L'.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -k, the k-th argument had an illegal value
*          > 0: if INFO = k, the leading minor of order k is not
*               positive definite, and the factorization could not be
*               completed.
*

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

```
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## cpotrf

```USAGE:
info, a = NumRu::Lapack.cpotrf( uplo, a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOTRF( UPLO, N, A, LDA, INFO )

*  Purpose
*  =======
*
*  CPOTRF computes the Cholesky factorization of a complex Hermitian
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U**H * U,  if UPLO = 'U', or
*     A = L  * L**H,  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the block version of the algorithm, calling Level 3 BLAS.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= 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, the leading minor of order i is not
*                positive definite, and the factorization could not be
*                completed.
*

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

```
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## cpotri

```USAGE:
info, a = NumRu::Lapack.cpotri( uplo, a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOTRI( UPLO, N, A, LDA, INFO )

*  Purpose
*  =======
*
*  CPOTRI computes the inverse of a complex Hermitian positive definite
*  matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
*  computed by CPOTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the triangular factor U or L from the Cholesky
*          factorization A = U**H*U or A = L*L**H, as computed by
*          CPOTRF.
*          On exit, the upper or lower triangle of the (Hermitian)
*          inverse of A, overwriting the input factor U or L.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= 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, the (i,i) element of the factor U or L is
*                zero, and the inverse could not be computed.
*

*  =====================================================================
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           CLAUUM, CTRTRI, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## cpotrs

```USAGE:
info, b = NumRu::Lapack.cpotrs( uplo, a, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )

*  Purpose
*  =======
*
*  CPOTRS solves a system of linear equations A*X = B with a Hermitian
*  positive definite matrix A using the Cholesky factorization
*  A = U**H*U or A = L*L**H computed by CPOTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**H*U or A = L*L**H, as computed by CPOTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  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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