# COMPLEX routines for symmetric matrix

## csycon

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

FORTRAN MANUAL
SUBROUTINE CSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO )

*  Purpose
*  =======
*
*  CSYCON estimates the reciprocal of the condition number (in the
*  1-norm) of a complex symmetric matrix A using the factorization
*  A = U*D*U**T or A = L*D*L**T computed by CSYTRF.
*
*  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
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by CSYTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
*  ANORM   (input) REAL
*          The 1-norm of the original 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)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

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

```
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## csyconv

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

FORTRAN MANUAL
SUBROUTINE CSYCONV( UPLO, WAY, N, A, LDA, IPIV, WORK, INFO )

*  Purpose
*  =======
*
*  CSYCONV convert A given by TRF into L and D and vice-versa.
*  Get Non-diag elements of D (returned in workspace) and
*  apply or reverse permutation done in TRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  WAY     (input) CHARACTER*1
*          = 'C': Convert
*          = 'R': Revert
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by CSYTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
* WORK     (workspace) COMPLEX array, dimension (N)
*
* LWORK    (input) INTEGER
*          The length of WORK.  LWORK >=1.
*          LWORK = N
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

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

```
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## csyequb

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

FORTRAN MANUAL
SUBROUTINE CSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )

*  Purpose
*  =======
*
*  CSYEQUB computes row and column scalings intended to equilibrate a
*  symmetric 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
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input) COMPLEX array, dimension (LDA,N)
*          The N-by-N symmetric 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.
*
*  WORK    (workspace) COMPLEX array, dimension (3*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-th diagonal element is nonpositive.
*

*  Further Details
*  ======= =======
*
*  Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
*  Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
*  DOI 10.1023/B:NUMA.0000016606.32820.69
*
*  =====================================================================
*

```
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## csymv

```USAGE:
y = NumRu::Lapack.csymv( uplo, alpha, a, x, incx, beta, y, incy, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYMV( UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY )

*  Purpose
*  =======
*
*  CSYMV  performs the matrix-vector  operation
*
*     y := alpha*A*x + beta*y,
*
*  where alpha and beta are scalars, x and y are n element vectors and
*  A is an n by n symmetric matrix.
*

*  Arguments
*  ==========
*
*  UPLO     (input) CHARACTER*1
*           On entry, UPLO specifies whether the upper or lower
*           triangular part of the array A is to be referenced as
*           follows:
*
*              UPLO = 'U' or 'u'   Only the upper triangular part of A
*                                  is to be referenced.
*
*              UPLO = 'L' or 'l'   Only the lower triangular part of A
*                                  is to be referenced.
*
*           Unchanged on exit.
*
*  N        (input) INTEGER
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  ALPHA    (input) COMPLEX
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A        (input) COMPLEX array, dimension ( LDA, N )
*           Before entry, with  UPLO = 'U' or 'u', the leading n by n
*           upper triangular part of the array A must contain the upper
*           triangular part of the symmetric matrix and the strictly
*           lower triangular part of A is not referenced.
*           Before entry, with UPLO = 'L' or 'l', the leading n by n
*           lower triangular part of the array A must contain the lower
*           triangular part of the symmetric matrix and the strictly
*           upper triangular part of A is not referenced.
*           Unchanged on exit.
*
*  LDA      (input) INTEGER
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, N ).
*           Unchanged on exit.
*
*  X        (input) COMPLEX array, dimension at least
*           ( 1 + ( N - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the N-
*           element vector x.
*           Unchanged on exit.
*
*  INCX     (input) INTEGER
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*  BETA     (input) COMPLEX
*           On entry, BETA specifies the scalar beta. When BETA is
*           supplied as zero then Y need not be set on input.
*           Unchanged on exit.
*
*  Y        (input/output) COMPLEX array, dimension at least
*           ( 1 + ( N - 1 )*abs( INCY ) ).
*           Before entry, the incremented array Y must contain the n
*           element vector y. On exit, Y is overwritten by the updated
*           vector y.
*
*  INCY     (input) INTEGER
*           On entry, INCY specifies the increment for the elements of
*           Y. INCY must not be zero.
*           Unchanged on exit.
*

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

```
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## csyr

```USAGE:
a = NumRu::Lapack.csyr( uplo, alpha, x, incx, a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYR( UPLO, N, ALPHA, X, INCX, A, LDA )

*  Purpose
*  =======
*
*  CSYR   performs the symmetric rank 1 operation
*
*     A := alpha*x*( x' ) + A,
*
*  where alpha is a complex scalar, x is an n element vector and A is an
*  n by n symmetric matrix.
*

*  Arguments
*  ==========
*
*  UPLO     (input) CHARACTER*1
*           On entry, UPLO specifies whether the upper or lower
*           triangular part of the array A is to be referenced as
*           follows:
*
*              UPLO = 'U' or 'u'   Only the upper triangular part of A
*                                  is to be referenced.
*
*              UPLO = 'L' or 'l'   Only the lower triangular part of A
*                                  is to be referenced.
*
*           Unchanged on exit.
*
*  N        (input) INTEGER
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  ALPHA    (input) COMPLEX
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  X        (input) COMPLEX array, dimension at least
*           ( 1 + ( N - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the N-
*           element vector x.
*           Unchanged on exit.
*
*  INCX     (input) INTEGER
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*  A        (input/output) COMPLEX array, dimension ( LDA, N )
*           Before entry, with  UPLO = 'U' or 'u', the leading n by n
*           upper triangular part of the array A must contain the upper
*           triangular part of the symmetric matrix and the strictly
*           lower triangular part of A is not referenced. On exit, the
*           upper triangular part of the array A is overwritten by the
*           upper triangular part of the updated matrix.
*           Before entry, with UPLO = 'L' or 'l', the leading n by n
*           lower triangular part of the array A must contain the lower
*           triangular part of the symmetric matrix and the strictly
*           upper triangular part of A is not referenced. On exit, the
*           lower triangular part of the array A is overwritten by the
*           lower triangular part of the updated matrix.
*
*  LDA      (input) INTEGER
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, N ).
*           Unchanged on exit.
*

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

```
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## csyrfs

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

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

*  Purpose
*  =======
*
*  CSYRFS improves the computed solution to a system of linear
*  equations when the coefficient matrix is symmetric indefinite, 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 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 factored form of the matrix A.  AF contains the block
*          diagonal matrix D and the multipliers used to obtain the
*          factor U or L from the factorization A = U*D*U**T or
*          A = L*D*L**T as computed by CSYTRF.
*
*  LDAF    (input) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
*  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 CSYTRS.
*          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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## csyrfsx

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

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

*     Purpose
*     =======
*
*     CSYRFSX improves the computed solution to a system of linear
*     equations when the coefficient matrix is symmetric indefinite, 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 factored form of the matrix A.  AF contains the block
*     diagonal matrix D and the multipliers used to obtain the
*     factor U or L from the factorization A = U*D*U**T or A =
*     L*D*L**T as computed by SSYTRF.
*
*     LDAF    (input) INTEGER
*     The leading dimension of the array AF.  LDAF >= max(1,N).
*
*     IPIV    (input) INTEGER array, dimension (N)
*     Details of the interchanges and the block structure of D
*     as determined by SSYTRF.
*
*     S       (input or output) REAL array, dimension (N)
*     The 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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## csysv

```USAGE:
ipiv, work, info, a, b = NumRu::Lapack.csysv( uplo, a, b, [:lwork => lwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO )

*  Purpose
*  =======
*
*  CSYSV computes the solution to a complex system of linear equations
*     A * X = B,
*  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*  matrices.
*
*  The diagonal pivoting method is used to factor A as
*     A = U * D * U**T,  if UPLO = 'U', or
*     A = L * D * L**T,  if UPLO = 'L',
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, and D is symmetric and block diagonal with
*  1-by-1 and 2-by-2 diagonal blocks.  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 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.
*
*          On exit, if INFO = 0, the block diagonal matrix D and the
*          multipliers used to obtain the factor U or L from the
*          factorization A = U*D*U**T or A = L*D*L**T as computed by
*          CSYTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D, as
*          determined by CSYTRF.  If IPIV(k) > 0, then rows and columns
*          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*          then rows and columns k-1 and -IPIV(k) were interchanged and
*          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
*          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*          diagonal block.
*
*  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).
*
*  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of WORK.  LWORK >= 1, and for best performance
*          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
*          CSYTRF.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
*               has been completed, but the block diagonal matrix D is
*               exactly singular, so the solution could not be computed.
*

*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            LQUERY
INTEGER            LWKOPT, NB
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
EXTERNAL           ILAENV, LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           CSYTRF, CSYTRS2, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## csysvx

```USAGE:
x, rcond, ferr, berr, work, info, af, ipiv = NumRu::Lapack.csysvx( fact, uplo, a, af, ipiv, b, [:lwork => lwork, :usage => usage, :help => help])

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

*  Purpose
*  =======
*
*  CSYSVX uses the diagonal pivoting factorization to compute the
*  solution to a complex system of linear equations A * X = B,
*  where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A.
*     The form of the factorization is
*        A = U * D * U**T,  if UPLO = 'U', or
*        A = L * D * L**T,  if UPLO = 'L',
*     where U (or L) is a product of permutation and unit upper (lower)
*     triangular matrices, and D is symmetric and block diagonal with
*     1-by-1 and 2-by-2 diagonal blocks.
*
*  2. If some D(i,i)=0, so that D 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.
*
*  3. The system of equations is solved for X using the factored form
*     of A.
*
*  4. Iterative refinement is applied to improve the computed solution
*     matrix and calculate error bounds and backward error estimates
*     for it.
*

*  Arguments
*  =========
*
*  FACT    (input) CHARACTER*1
*          Specifies whether or not the factored form of A has been
*          supplied on entry.
*          = 'F':  On entry, AF and IPIV contain the factored form
*                  of A.  A, AF and IPIV will not be modified.
*          = 'N':  The matrix A will be 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) 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 or output) COMPLEX array, dimension (LDAF,N)
*          If FACT = 'F', then AF is an input argument and on entry
*          contains the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L from the factorization
*          A = U*D*U**T or A = L*D*L**T as computed by CSYTRF.
*
*          If FACT = 'N', then AF is an output argument and on exit
*          returns the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L from the factorization
*          A = U*D*U**T or A = L*D*L**T.
*
*  LDAF    (input) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  IPIV    (input or output) INTEGER array, dimension (N)
*          If FACT = 'F', then IPIV is an input argument and on entry
*          contains details of the interchanges and the block structure
*          of D, as determined by CSYTRF.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*          If FACT = 'N', then IPIV is an output argument and on exit
*          contains details of the interchanges and the block structure
*          of D, as determined by CSYTRF.
*
*  B       (input) COMPLEX array, dimension (LDB,NRHS)
*          The N-by-NRHS right hand side matrix 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.
*
*  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.  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/output) COMPLEX array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of WORK.  LWORK >= max(1,2*N), and for best
*          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where
*          NB is the optimal blocksize for CSYTRF.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  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:  D(i,i) is exactly zero.  The factorization
*                       has been completed but the factor D is exactly
*                       singular, so the solution and error bounds could
*                       not be computed. RCOND = 0 is returned.
*                = N+1: D 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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## csysvxx

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

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

*     Purpose
*     =======
*
*     CSYSVXX uses the diagonal pivoting factorization to compute the
*     solution to a complex system of linear equations A * X = B, where
*     A is an N-by-N symmetric matrix and X and B are N-by-NRHS
*     matrices.
*
*     If requested, both normwise and maximum componentwise error bounds
*     are returned. CSYSVXX 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.
*
*     CSYSVXX 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
*     CSYSVXX 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 CSYSVXX 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 LU decomposition is used to factor
*     the matrix A (after equilibration if FACT = 'E') as
*
*        A = U * D * U**T,  if UPLO = 'U', or
*        A = L * D * L**T,  if UPLO = 'L',
*
*     where U (or L) is a product of permutation and unit upper (lower)
*     triangular matrices, and D is symmetric and block diagonal with
*     1-by-1 and 2-by-2 diagonal blocks.
*
*     3. If some D(i,i)=0, so that D 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(R) 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 S.
*               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.
*
*     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)
*     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.
*
*     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 block diagonal matrix D and the multipliers
*     used to obtain the factor U or L from the factorization A =
*     U*D*U**T or A = L*D*L**T as computed by SSYTRF.
*
*     If FACT = 'N', then AF is an output argument and on exit
*     returns the block diagonal matrix D and the multipliers
*     used to obtain the factor U or L from the factorization A =
*     U*D*U**T or A = L*D*L**T.
*
*     LDAF    (input) INTEGER
*     The leading dimension of the array AF.  LDAF >= max(1,N).
*
*     IPIV    (input or output) INTEGER array, dimension (N)
*     If FACT = 'F', then IPIV is an input argument and on entry
*     contains details of the interchanges and the block
*     structure of D, as determined by SSYTRF.  If IPIV(k) > 0,
*     then rows and columns k and IPIV(k) were interchanged and
*     D(k,k) is a 1-by-1 diagonal block.  If UPLO = 'U' and
*     IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
*     -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
*     diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
*     then rows and columns k+1 and -IPIV(k) were interchanged
*     and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*     If FACT = 'N', then IPIV is an output argument and on exit
*     contains details of the interchanges and the block
*     structure of D, as determined by SSYTRF.
*
*     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 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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## csyswapr

```USAGE:
a = NumRu::Lapack.csyswapr( uplo, a, i1, i2, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYSWAPR( UPLO, N, A, I1, I2)

*  Purpose
*  =======
*
*  CSYSWAPR applies an elementary permutation on the rows and the columns of
*  a symmetric matrix.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the NB diagonal matrix D and the multipliers
*          used to obtain the factor U or L as computed by CSYTRF.
*
*          On exit, if INFO = 0, the (symmetric) inverse of the original
*          matrix.  If UPLO = 'U', the upper triangular part of the
*          inverse is formed and the part of A below the diagonal is not
*          referenced; if UPLO = 'L' the lower triangular part of the
*          inverse is formed and the part of A above the diagonal is
*          not referenced.
*
*  I1      (input) INTEGER
*          Index of the first row to swap
*
*  I2      (input) INTEGER
*          Index of the second row to swap
*

*  =====================================================================
*
*     ..
*     .. Local Scalars ..
LOGICAL            UPPER
INTEGER            I
COMPLEX            TMP
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           CSWAP
*     ..

```
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## csytf2

```USAGE:
ipiv, info, a = NumRu::Lapack.csytf2( uplo, a, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYTF2( UPLO, N, A, LDA, IPIV, INFO )

*  Purpose
*  =======
*
*  CSYTF2 computes the factorization of a complex symmetric matrix A
*  using the Bunch-Kaufman diagonal pivoting method:
*
*     A = U*D*U'  or  A = L*D*L'
*
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, U' is the transpose of U, and D is symmetric and
*  block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*
*  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
*          symmetric 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 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.
*
*          On exit, the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L (see below for further details).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -k, the k-th argument had an illegal value
*          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
*               has been completed, but the block diagonal matrix D is
*               exactly singular, and division by zero will occur if it
*               is used to solve a system of equations.
*

*  Further Details
*  ===============
*
*  09-29-06 - patch from
*    Bobby Cheng, MathWorks
*
*    Replace l.209 and l.377
*         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*    by
*         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
*
*  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
*         Company
*
*  If UPLO = 'U', then A = U*D*U', where
*     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    v    0   )   k-s
*     U(k) =  (   0    I    0   )   s
*             (   0    0    I   )   n-k
*                k-s   s   n-k
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*
*  If UPLO = 'L', then A = L*D*L', where
*     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    0     0   )  k-1
*     L(k) =  (   0    I     0   )  s
*             (   0    v     I   )  n-k-s+1
*                k-1   s  n-k-s+1
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*
*  =====================================================================
*

```
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## csytrf

```USAGE:
ipiv, work, info, a = NumRu::Lapack.csytrf( uplo, a, lwork, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )

*  Purpose
*  =======
*
*  CSYTRF computes the factorization of a complex symmetric matrix A
*  using the Bunch-Kaufman diagonal pivoting method.  The form of the
*  factorization is
*
*     A = U*D*U**T  or  A = L*D*L**T
*
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, and D is symmetric and block diagonal with
*  with 1-by-1 and 2-by-2 diagonal blocks.
*
*  This is the blocked 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 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.
*
*          On exit, the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L (see below for further details).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of WORK.  LWORK >=1.  For best performance
*          LWORK >= N*NB, where NB is the block size returned by ILAENV.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
*                has been completed, but the block diagonal matrix D is
*                exactly singular, and division by zero will occur if it
*                is used to solve a system of equations.
*

*  Further Details
*  ===============
*
*  If UPLO = 'U', then A = U*D*U', where
*     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    v    0   )   k-s
*     U(k) =  (   0    I    0   )   s
*             (   0    0    I   )   n-k
*                k-s   s   n-k
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*
*  If UPLO = 'L', then A = L*D*L', where
*     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    0     0   )  k-1
*     L(k) =  (   0    I     0   )  s
*             (   0    v     I   )  n-k-s+1
*                k-1   s  n-k-s+1
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*
*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            LQUERY, UPPER
INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
EXTERNAL           CLASYF, CSYTF2, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## csytri

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

FORTRAN MANUAL
SUBROUTINE CSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO )

*  Purpose
*  =======
*
*  CSYTRI computes the inverse of a complex symmetric indefinite matrix
*  A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*  CSYTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the block diagonal matrix D and the multipliers
*          used to obtain the factor U or L as computed by CSYTRF.
*
*          On exit, if INFO = 0, the (symmetric) inverse of the original
*          matrix.  If UPLO = 'U', the upper triangular part of the
*          inverse is formed and the part of A below the diagonal is not
*          referenced; if UPLO = 'L' the lower triangular part of the
*          inverse is formed and the part of A above the diagonal is
*          not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
*  WORK    (workspace) COMPLEX array, dimension (2*N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*               inverse could not be computed.
*

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

```
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## csytri2

```USAGE:
info, a = NumRu::Lapack.csytri2( uplo, a, ipiv, [:lwork => lwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYTRI2( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )

*  Purpose
*  =======
*
*  CSYTRI2 computes the inverse of a complex symmetric indefinite matrix
*  A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*  CSYTRF. CSYTRI2 sets the LEADING DIMENSION of the workspace
*  before calling CSYTRI2X that actually computes the inverse.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the NB diagonal matrix D and the multipliers
*          used to obtain the factor U or L as computed by CSYTRF.
*
*          On exit, if INFO = 0, the (symmetric) inverse of the original
*          matrix.  If UPLO = 'U', the upper triangular part of the
*          inverse is formed and the part of A below the diagonal is not
*          referenced; if UPLO = 'L' the lower triangular part of the
*          inverse is formed and the part of A above the diagonal is
*          not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the NB structure of D
*          as determined by CSYTRF.
*
*  WORK    (workspace) COMPLEX array, dimension (N+NB+1)*(NB+3)
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.
*          WORK is size >= (N+NB+1)*(NB+3)
*          If LDWORK = -1, then a workspace query is assumed; the routine
*           calculates:
*              - the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array,
*              - and no error message related to LDWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*               inverse could not be computed.
*

*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            UPPER, LQUERY
INTEGER            MINSIZE, NBMAX
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
EXTERNAL           CSYTRI2X
*     ..

```
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## csytri2x

```USAGE:
info, a = NumRu::Lapack.csytri2x( uplo, a, ipiv, nb, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )

*  Purpose
*  =======
*
*  CSYTRI2X computes the inverse of a real symmetric indefinite matrix
*  A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*  CSYTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) COMPLEX array, dimension (LDA,N)
*          On entry, the NNB diagonal matrix D and the multipliers
*          used to obtain the factor U or L as computed by CSYTRF.
*
*          On exit, if INFO = 0, the (symmetric) inverse of the original
*          matrix.  If UPLO = 'U', the upper triangular part of the
*          inverse is formed and the part of A below the diagonal is not
*          referenced; if UPLO = 'L' the lower triangular part of the
*          inverse is formed and the part of A above the diagonal is
*          not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the NNB structure of D
*          as determined by CSYTRF.
*
*  WORK    (workspace) COMPLEX array, dimension (N+NNB+1,NNB+3)
*
*  NB      (input) INTEGER
*          Block size
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*               inverse could not be computed.
*

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

```
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## csytrs

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

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

*  Purpose
*  =======
*
*  CSYTRS solves a system of linear equations A*X = B with a complex
*  symmetric matrix A using the factorization A = U*D*U**T or
*  A = L*D*L**T computed by CSYTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  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 block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by CSYTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
*  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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## csytrs2

```USAGE:
info, b = NumRu::Lapack.csytrs2( uplo, a, ipiv, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CSYTRS2( UPLO, N, NRHS, A, LDA, IPIV, B, LDB,  WORK, INFO )

*  Purpose
*  =======
*
*  CSYTRS2 solves a system of linear equations A*X = B with a COMPLEX
*  symmetric matrix A using the factorization A = U*D*U**T or
*  A = L*D*L**T computed by CSYTRF and converted by CSYCONV.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  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 block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by CSYTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by CSYTRF.
*
*  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).
*
*  WORK    (workspace) COMPLEX 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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