CHESV computes the solution to a complex system of linear equations


    A * X = B,


 where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS


 matrices.


 The diagonal pivoting method is used to factor A as


    A = U * D * U**H,  if UPLO = 'U', or


    A = L * D * L**H,  if UPLO = 'L',


 where U (or L) is a product of permutation and unit upper (lower)


 triangular matrices, and D is Hermitian 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

A

          A is 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 block diagonal matrix D and the


          multipliers used to obtain the factor U or L from the


          factorization A = U*D*U**H or A = L*D*L**H as computed by


          CHETRF.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

IPIV

          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D, as


          determined by CHETRF.  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

          B is 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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= 1, and for best performance


          LWORK >= max(1,N*NB), where NB is the optimal blocksize for


          CHETRF.


          for LWORK < N, TRS will be done with Level BLAS 2


          for LWORK >= N, TRS will be done with Level BLAS 3


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CHESV_AA computes the solution to a complex system of linear equations


    A * X = B,


 where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS


 matrices.


 Aasen's algorithm is used to factor A as


    A = U**H * T * U,  if UPLO = 'U', or


    A = L * T * L**H,  if UPLO = 'L',


 where U (or L) is a product of permutation and unit upper (lower)


 triangular matrices, and T is Hermitian and tridiagonal. The factored form


 of A is then used to solve the system of equations A * X = B.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

A

          A is 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 tridiagonal matrix T and the


          multipliers used to obtain the factor U or L from the


          factorization A = U**H*T*U or A = L*T*L**H as computed by


          CHETRF_AA.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

IPIV

          IPIV is INTEGER array, dimension (N)


          On exit, it contains the details of the interchanges, i.e.,


          the row and column k of A were interchanged with the


          row and column IPIV(k).

B

          B is 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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= MAX(1,2*N,3*N-2), and for best 


          performance LWORK >= MAX(1,N*NB), where NB is the optimal


          blocksize for CHETRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CHESV_RK computes the solution to a complex system of linear


 equations A * X = B, where A is an N-by-N Hermitian matrix


 and X and B are N-by-NRHS matrices.


 The bounded Bunch-Kaufman (rook) diagonal pivoting method is used


 to factor A as


    A = P*U*D*(U**H)*(P**T),  if UPLO = 'U', or


    A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',


 where U (or L) is unit upper (or lower) triangular matrix,


 U**H (or L**H) is the conjugate of U (or L), P is a permutation


 matrix, P**T is the transpose of P, and D is Hermitian and block


 diagonal with 1-by-1 and 2-by-2 diagonal blocks.


 CHETRF_RK is called to compute the factorization of a complex


 Hermitian matrix.  The factored form of A is then used to solve


 the system of equations A * X = B by calling BLAS3 routine CHETRS_3.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          Hermitian matrix A is stored:


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

A

          A is 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, diagonal of the block diagonal


          matrix D and factors U or L  as computed by CHETRF_RK:


            a) ONLY diagonal elements of the Hermitian block diagonal


               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);


               (superdiagonal (or subdiagonal) elements of D


                are stored on exit in array E), and


            b) If UPLO = 'U': factor U in the superdiagonal part of A.


               If UPLO = 'L': factor L in the subdiagonal part of A.


          For more info see the description of CHETRF_RK routine.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

E

          E is COMPLEX array, dimension (N)


          On exit, contains the output computed by the factorization


          routine CHETRF_RK, i.e. the superdiagonal (or subdiagonal)


          elements of the Hermitian block diagonal matrix D


          with 1-by-1 or 2-by-2 diagonal blocks, where


          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;


          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.


          NOTE: For 1-by-1 diagonal block D(k), where


          1 <= k <= N, the element E(k) is set to 0 in both


          UPLO = 'U' or UPLO = 'L' cases.


          For more info see the description of CHETRF_RK routine.

IPIV

          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D,


          as determined by CHETRF_RK.


          For more info see the description of CHETRF_RK routine.

B

          B is 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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

WORK

          WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).


          Work array used in the factorization stage.


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= 1. For best performance


          of factorization stage LWORK >= max(1,N*NB), where NB is


          the optimal blocksize for CHETRF_RK.


          If LWORK = -1, then a workspace query is assumed;


          the routine only calculates the optimal size of the WORK


          array for factorization stage, returns this value as


          the first entry of the WORK array, and no error message


          related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: If INFO = -k, the k-th argument had an illegal value


          > 0: If INFO = k, the matrix A is singular, because:


                 If UPLO = 'U': column k in the upper


                 triangular part of A contains all zeros.


                 If UPLO = 'L': column k in the lower


                 triangular part of A contains all zeros.


               Therefore D(k,k) is exactly zero, and superdiagonal


               elements of column k of U (or subdiagonal elements of


               column k of L ) are all zeros. 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.


               NOTE: INFO only stores the first occurrence of


               a singularity, any subsequent occurrence of singularity


               is not stored in INFO even though the factorization


               always completes.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  December 2016,  Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley


  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,


                  School of Mathematics,


                  University of Manchester

 CHESV_ROOK computes the solution to a complex system of linear equations


    A * X = B,


 where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS


 matrices.


 The bounded Bunch-Kaufman ('rook') 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 Hermitian and block diagonal with


 1-by-1 and 2-by-2 diagonal blocks.


 CHETRF_ROOK is called to compute the factorization of a complex


 Hermition matrix A using the bounded Bunch-Kaufman ('rook') diagonal


 pivoting method.


 The factored form of A is then used to solve the system


 of equations A * X = B by calling CHETRS_ROOK (uses BLAS 2).

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

A

          A is 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 block diagonal matrix D and the


          multipliers used to obtain the factor U or L from the


          factorization A = U*D*U**H or A = L*D*L**H as computed by


          CHETRF_ROOK.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

IPIV

          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D.


          If UPLO = 'U':


             Only the last KB elements of IPIV are set.


             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 IPIV(k) < 0 and IPIV(k-1) < 0, then rows and


             columns k and -IPIV(k) were interchanged and rows and


             columns k-1 and -IPIV(k-1) were inerchaged,


             D(k-1:k,k-1:k) is a 2-by-2 diagonal block.


          If UPLO = 'L':


             Only the first KB elements of IPIV are set.


             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 IPIV(k) < 0 and IPIV(k+1) < 0, then rows and


             columns k and -IPIV(k) were interchanged and rows and


             columns k+1 and -IPIV(k+1) were inerchaged,


             D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

B

          B is 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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= 1, and for best performance


          LWORK >= max(1,N*NB), where NB is the optimal blocksize for


          CHETRF_ROOK.


          for LWORK < N, TRS will be done with Level BLAS 2


          for LWORK >= N, TRS will be done with Level BLAS 3


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  November 2013,  Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley


  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,


                  School of Mathematics,


                  University of Manchester.fi

 CHESVX 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 Hermitian matrix and X and B are N-by-NRHS


 matrices.


 Error bounds on the solution and a condition estimate are also


 provided.

 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**H,  if UPLO = 'U', or


       A = L * D * L**H,  if UPLO = 'L',


    where U (or L) is a product of permutation and unit upper (lower)


    triangular matrices, and D is Hermitian 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.

FACT

          FACT is 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

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

A

          A is 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

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is 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**H or A = L*D*L**H as computed by CHETRF.


          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**H or A = L*D*L**H.

LDAF

          LDAF is INTEGER


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

IPIV

          IPIV is 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 CHETRF.


          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 CHETRF.

B

          B is COMPLEX array, dimension (LDB,NRHS)


          The N-by-NRHS right hand side matrix B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

          LDX is INTEGER


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

RCOND

          RCOND is 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

          FERR is REAL array, dimension (NRHS)


          The estimated forward error bound for each solution vector


          X(j) (the j-th column of the solution matrix X).


          If XTRUE is the true solution corresponding to X(j), FERR(j)


          is an estimated upper bound for the magnitude of the largest


          element in (X(j) - XTRUE) divided by the magnitude of the


          largest element in X(j).  The estimate is as reliable as


          the estimate for RCOND, and is almost always a slight


          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)


          The componentwise relative backward error of each solution


          vector X(j) (i.e., the smallest relative change in


          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is 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 CHETRF.


          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

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

    CHESVXX 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 Hermitian matrix and X and B are N-by-NRHS


    matrices.


    If requested, both normwise and maximum componentwise error bounds


    are returned. CHESVXX 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.


    CHESVXX 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


    CHESVXX 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 CHESVXX would itself produce.

    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 Hermitian 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.

     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

          FACT is 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

          UPLO is CHARACTER*1


       = 'U':  Upper triangle of A is stored;


       = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


     The number of linear equations, i.e., the order of the


     matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


     The number of right hand sides, i.e., the number of columns


     of the matrices B and X.  NRHS >= 0.

A

          A is 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.


     On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by


     diag(S)*A*diag(S).

LDA

          LDA is INTEGER


     The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is 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**H or A = L*D*L**H as computed by CHETRF.


     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**H or A = L*D*L**H.

LDAF

          LDAF is INTEGER


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

IPIV

          IPIV is 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 CHETRF.  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 CHETRF.

EQUED

          EQUED is 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

          S is 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

          B is 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

          LDB is INTEGER


     The leading dimension of the array B.  LDB >= max(1,N).

X

          X is 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

          LDX is INTEGER


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

RCOND

          RCOND is REAL


     Reciprocal scaled condition number.  This is an estimate of the


     reciprocal Skeel condition number of the matrix A after


     equilibration (if done).  If this is less than the machine


     precision (in particular, if it is zero), the matrix is singular


     to working precision.  Note that the error may still be small even


     if this number is very small and the matrix appears ill-


     conditioned.

RPVGRW

          RPVGRW is 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<INFO<=N, then this contains the reciprocal pivot growth factor


     for the leading INFO columns of A.

BERR

          BERR is REAL array, dimension (NRHS)


     Componentwise relative backward error.  This is the


     componentwise relative backward error of each solution vector X(j)


     (i.e., the smallest relative change in any element of A or B that


     makes X(j) an exact solution).

N_ERR_BNDS

          N_ERR_BNDS is INTEGER


     Number of error bounds to return for each right hand side


     and each type (normwise or componentwise).  See ERR_BNDS_NORM and


     ERR_BNDS_COMP below.

ERR_BNDS_NORM

          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)


     For each right-hand side, this array contains information about


     various error bounds and condition numbers corresponding to the


     normwise relative error, which is defined as follows:


     Normwise relative error in the ith solution vector:


             max_j (abs(XTRUE(j,i) - X(j,i)))


            ------------------------------


                  max_j abs(X(j,i))


     The array is indexed by the type of error information as described


     below. There currently are up to three pieces of information


     returned.


     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith


     right-hand side.


     The second index in ERR_BNDS_NORM(:,err) contains the following


     three fields:


     err = 1 'Trust/don't trust' boolean. Trust the answer if the


              reciprocal condition number is less than the threshold


              sqrt(n) * slamch('Epsilon').


     err = 2 'Guaranteed' error bound: The estimated forward error,


              almost certainly within a factor of 10 of the true error


              so long as the next entry is greater than the threshold


              sqrt(n) * slamch('Epsilon'). This error bound should only


              be trusted if the previous boolean is true.


     err = 3  Reciprocal condition number: Estimated normwise


              reciprocal condition number.  Compared with the threshold


              sqrt(n) * slamch('Epsilon') to determine if the error


              estimate is 'guaranteed'. These reciprocal condition


              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some


              appropriately scaled matrix Z.


              Let Z = S*A, where S scales each row by a power of the


              radix so all absolute row sums of Z are approximately 1.


     See Lapack Working Note 165 for further details and extra


     cautions.

ERR_BNDS_COMP

          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)


     For each right-hand side, this array contains information about


     various error bounds and condition numbers corresponding to the


     componentwise relative error, which is defined as follows:


     Componentwise relative error in the ith solution vector:


                    abs(XTRUE(j,i) - X(j,i))


             max_j ----------------------


                         abs(X(j,i))


     The array is indexed by the right-hand side i (on which the


     componentwise relative error depends), and the type of error


     information as described below. There currently are up to three


     pieces of information returned for each right-hand side. If


     componentwise accuracy is not requested (PARAMS(3) = 0.0), then


     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most


     the first (:,N_ERR_BNDS) entries are returned.


     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith


     right-hand side.


     The second index in ERR_BNDS_COMP(:,err) contains the following


     three fields:


     err = 1 'Trust/don't trust' boolean. Trust the answer if the


              reciprocal condition number is less than the threshold


              sqrt(n) * slamch('Epsilon').


     err = 2 'Guaranteed' error bound: The estimated forward error,


              almost certainly within a factor of 10 of the true error


              so long as the next entry is greater than the threshold


              sqrt(n) * slamch('Epsilon'). This error bound should only


              be trusted if the previous boolean is true.


     err = 3  Reciprocal condition number: Estimated componentwise


              reciprocal condition number.  Compared with the threshold


              sqrt(n) * slamch('Epsilon') to determine if the error


              estimate is 'guaranteed'. These reciprocal condition


              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some


              appropriately scaled matrix Z.


              Let Z = S*(A*diag(x)), where x is the solution for the


              current right-hand side and S scales each row of


              A*diag(x) by a power of the radix so all absolute row


              sums of Z are approximately 1.


     See Lapack Working Note 165 for further details and extra


     cautions.

NPARAMS

          NPARAMS is INTEGER


     Specifies the number of parameters set in PARAMS.  If <= 0, the


     PARAMS array is never referenced and default values are used.

PARAMS

          PARAMS is REAL array, dimension NPARAMS


     Specifies algorithm parameters.  If an entry is < 0.0, then


     that entry will be filled with default value used for that


     parameter.  Only positions up to NPARAMS are accessed; defaults


     are used for higher-numbered parameters.


       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative


            refinement or not.


         Default: 1.0


            = 0.0:  No refinement is performed, and no error bounds are


                    computed.


            = 1.0:  Use the double-precision refinement algorithm,


                    possibly with doubled-single computations if the


                    compilation environment does not support DOUBLE


                    PRECISION.


              (other values are reserved for future use)


       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual


            computations allowed for refinement.


         Default: 10


         Aggressive: Set to 100 to permit convergence using approximate


                     factorizations or factorizations other than LU. If


                     the factorization uses a technique other than


                     Gaussian elimination, the guarantees in


                     err_bnds_norm and err_bnds_comp may no longer be


                     trustworthy.


       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code


            will attempt to find a solution with small componentwise


            relative error in the double-precision algorithm.  Positive


            is true, 0.0 is false.


         Default: 1.0 (attempt componentwise convergence)

WORK

          WORK is COMPLEX array, dimension (5*N)

RWORK

          RWORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER


       = 0:  Successful exit. The solution to every right-hand side is


         guaranteed.


       < 0:  If INFO = -i, the i-th argument had an illegal value


       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization


         has been completed, but the factor U is exactly singular, so


         the solution and error bounds could not be computed. RCOND = 0


         is returned.


       = N+J: The solution corresponding to the Jth right-hand side is


         not guaranteed. The solutions corresponding to other right-


         hand sides K with K > J may not be guaranteed as well, but


         only the first such right-hand side is reported. If a small


         componentwise error is not requested (PARAMS(3) = 0.0) then


         the Jth right-hand side is the first with a normwise error


         bound that is not guaranteed (the smallest J such


         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)


         the Jth right-hand side is the first with either a normwise or


         componentwise error bound that is not guaranteed (the smallest


         J such that either ERR_BNDS_NORM(J,1) = 0.0 or


         ERR_BNDS_COMP(J,1) = 0.0). See the definition of


         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information


         about all of the right-hand sides check ERR_BNDS_NORM or


         ERR_BNDS_COMP.

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