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NAME

       complexHEauxiliary

SYNOPSIS

   Functions
       subroutine cheswapr (UPLO, N, A, LDA, I1, I2)
           CHESWAPR applies an elementary permutation on the rows and columns of a Hermitian
           matrix.
       real function clanhe (NORM, UPLO, N, A, LDA, WORK)
           CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a complex Hermitian matrix.
       subroutine claqhe (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
           CLAQHE scales a Hermitian matrix.

Detailed Description

       This is the group of complex auxiliary functions for HE matrices

Function Documentation

   subroutine cheswapr (character UPLO, integer N, complex, dimension( lda, n ) A, integer LDA,
       integer I1, integer I2)
       CHESWAPR applies an elementary permutation on the rows and columns of a Hermitian matrix.

       Purpose:

            CHESWAPR applies an elementary permutation on the rows and the columns of
            a hermitian matrix.

       Parameters:
           UPLO

                     UPLO is 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

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is 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

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

           I1

                     I1 is INTEGER
                     Index of the first row to swap

           I2

                     I2 is INTEGER
                     Index of the second row to swap

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   real function clanhe (character NORM, character UPLO, integer N, complex, dimension( lda, * )
       A, integer LDA, real, dimension( * ) WORK)
       CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a complex Hermitian matrix.

       Purpose:

            CLANHE  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the  element of  largest absolute value  of a
            complex hermitian matrix A.

       Returns:
           CLANHE

               CLANHE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters:
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in CLANHE as described
                     above.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     hermitian matrix A is to be referenced.
                     = 'U':  Upper triangular part of A is referenced
                     = 'L':  Lower triangular part of A is referenced

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, CLANHE is
                     set to zero.

           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. Note that the imaginary parts of the diagonal
                     elements need not be set and are assumed to be zero.

           LDA

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

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                     WORK is not referenced.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine claqhe (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) S, real SCOND, real AMAX, character EQUED)
       CLAQHE scales a Hermitian matrix.

       Purpose:

            CLAQHE equilibrates a Hermitian matrix A using the scaling factors
            in the vector S.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     Hermitian matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 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 EQUED = 'Y', the equilibrated matrix:
                     diag(S) * A * diag(S).

           LDA

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

           S

                     S is REAL array, dimension (N)
                     The scale factors for A.

           SCOND

                     SCOND is REAL
                     Ratio of the smallest S(i) to the largest S(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies whether or not equilibration was done.
                     = 'N':  No equilibration.
                     = 'Y':  Equilibration was done, i.e., A has been replaced by
                             diag(S) * A * diag(S).

       Internal Parameters:

             THRESH is a threshold value used to decide if scaling should be done
             based on the ratio of the scaling factors.  If SCOND < THRESH,
             scaling is done.

             LARGE and SMALL are threshold values used to decide if scaling should
             be done based on the absolute size of the largest matrix element.
             If AMAX > LARGE or AMAX < SMALL, scaling is done.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

Author

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