zlanhf.f - Man Page

double precision function zlanhf (NORM, TRANSR, UPLO, N, A, WORK)
ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.  

Purpose:

 ZLANHF  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 in RFP format.

Returns:

ZLANHF

    ZLANHF = ( 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
            Specifies the value to be returned in ZLANHF as described
            above.

TRANSR

          TRANSR is CHARACTER
            Specifies whether the RFP format of A is normal or
            conjugate-transposed format.
            = 'N':  RFP format is Normal
            = 'C':  RFP format is Conjugate-transposed

UPLO

          UPLO is CHARACTER
            On entry, UPLO specifies whether the RFP matrix A came from
            an upper or lower triangular matrix as follows:

            UPLO = 'U' or 'u' RFP A came from an upper triangular
            matrix

            UPLO = 'L' or 'l' RFP A came from a  lower triangular
            matrix

N

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

A

          A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
            On entry, the matrix A in RFP Format.
            RFP Format is described by TRANSR, UPLO and N as follows:
            If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
            K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
            TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A
            as defined when TRANSR = 'N'. The contents of RFP A are
            defined by UPLO as follows: If UPLO = 'U' the RFP A
            contains the ( N*(N+1)/2 ) elements of upper packed A
            either in normal or conjugate-transpose Format. If
            UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements
            of lower packed A either in normal or conjugate-transpose
            Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When
            TRANSR is 'N' the LDA is N+1 when N is even and is N when
            is odd. See the Note below for more details.
            Unchanged on exit.

WORK

          WORK is DOUBLE PRECISION array, dimension (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

Further Details:

  We first consider Standard Packed 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
  conjugate-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
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. 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 = 'C'. RFP A in both UPLO cases is just the conjugate-
  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 next  consider Standard Packed 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
  conjugate-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
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. 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 = 'C'. RFP A in both UPLO cases is just the conjugate-
  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 248 of file zlanhf.f.