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NAME

       dlansf.f -

SYNOPSIS

   Functions/Subroutines
       double precision function dlansf (NORM, TRANSR, UPLO, N, A, WORK)
           DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
           or the element of largest absolute value of a symmetric matrix in RFP format.

Function/Subroutine Documentation

   double precision function dlansf (characterNORM, characterTRANSR, characterUPLO, integerN,
       double precision, dimension( 0: * )A, double precision, dimension( 0: * )WORK)
       DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
       the element of largest absolute value of a symmetric matrix in RFP format.

       Purpose:

            DLANSF returns the value of the one norm, or the Frobenius norm, or
            the infinity norm, or the element of largest absolute value of a
            real symmetric matrix A in RFP format.

       Returns:
           DLANSF

               DLANSF = ( 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  matrix norm.

       Parameters:
           NORM

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

           TRANSR

                     TRANSR is CHARACTER*1
                     Specifies whether the RFP format of A is normal or
                     transposed format.
                     = 'N':  RFP format is Normal;
                     = 'T':  RFP format is Transpose.

           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the RFP matrix A came from
                      an upper or lower triangular matrix as follows:
                      = 'U': RFP A came from an upper triangular matrix;
                      = 'L': RFP A came from a lower triangular matrix.

           N

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

           A

                     A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
                     On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
                     part of the symmetric matrix A stored in RFP format. See the
                     "Notes" below for more details.
                     Unchanged on exit.

           WORK

                     WORK is DOUBLE PRECISION 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:
           September 2012

       Further Details:

             We first consider Rectangular Full Packed (RFP) Format when N is
             even. We give an example where N = 6.

                 AP is Upper             AP is Lower

              00 01 02 03 04 05       00
                 11 12 13 14 15       10 11
                    22 23 24 25       20 21 22
                       33 34 35       30 31 32 33
                          44 45       40 41 42 43 44
                             55       50 51 52 53 54 55

             Let TRANSR = 'N'. RFP holds AP as follows:
             For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
             three columns of AP upper. The lower triangle A(4:6,0:2) consists of
             the transpose of the first three columns of AP upper.
             For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
             three columns of AP lower. The upper triangle A(0:2,0:2) consists of
             the transpose of the last three columns of AP lower.
             This covers the case N even and TRANSR = 'N'.

                    RFP A                   RFP A

                   03 04 05                33 43 53
                   13 14 15                00 44 54
                   23 24 25                10 11 55
                   33 34 35                20 21 22
                   00 44 45                30 31 32
                   01 11 55                40 41 42
                   02 12 22                50 51 52

             Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
             transpose of RFP A above. One therefore gets:

                      RFP A                   RFP A

                03 13 23 33 00 01 02    33 00 10 20 30 40 50
                04 14 24 34 44 11 12    43 44 11 21 31 41 51
                05 15 25 35 45 55 22    53 54 55 22 32 42 52

             We then consider Rectangular Full Packed (RFP) Format when N is
             odd. We give an example where N = 5.

                AP is Upper                 AP is Lower

              00 01 02 03 04              00
                 11 12 13 14              10 11
                    22 23 24              20 21 22
                       33 34              30 31 32 33
                          44              40 41 42 43 44

             Let TRANSR = 'N'. RFP holds AP as follows:
             For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
             three columns of AP upper. The lower triangle A(3:4,0:1) consists of
             the transpose of the first two columns of AP upper.
             For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
             three columns of AP lower. The upper triangle A(0:1,1:2) consists of
             the transpose of the last two columns of AP lower.
             This covers the case N odd and TRANSR = 'N'.

                    RFP A                   RFP A

                   02 03 04                00 33 43
                   12 13 14                10 11 44
                   22 23 24                20 21 22
                   00 33 34                30 31 32
                   01 11 44                40 41 42

             Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
             transpose of RFP A above. One therefore gets:

                      RFP A                   RFP A

                02 12 22 00 01             00 10 20 30 40 50
                03 13 23 33 11             33 11 21 31 41 51
                04 14 24 34 44             43 44 22 32 42 52

       Definition at line 210 of file dlansf.f.

Author

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