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clanhf-f(3)

clanhf.f –

Synopsis

Functions/Subroutines

REAL function clanhf (NORM, TRANSR, UPLO, N, A, WORK)
CLANHF 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.

Function/Subroutine Documentation

REAL function clanhf (characterNORM, characterTRANSR, characterUPLO, integerN, complex, dimension( 0: * )A, real, dimension( 0: * )WORK)

CLANHF 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:

 CLANHF 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:

CLANHF

 CLANHF = ( 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 CLANHF 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, CLANHF is
 set to zero.

A

 A is COMPLEX 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 REAL 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:

September 2012

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 247 of file clanhf.f.

Author

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