CLANHF(1LAPACK routine (version 3.2)CLANHF(1)NAME
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 com‐
plex Hermitian matrix A in RFP format
SYNOPSIS
REAL FUNCTION CLANHF( NORM, TRANSR, UPLO, N, A, WORK )
CHARACTER NORM, TRANSR, UPLO
INTEGER N
REAL WORK( 0: * )
COMPLEX A( 0: * )
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.
DESCRIPTION
CLANHF returns the value
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.
ARGUMENTS
NORM (input) CHARACTER
Specifies the value to be returned in CLANHF as described
above.
TRANSR (input) 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 (input) 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 (input) INTEGER
The order of the matrix A. N >= 0. When N = 0, CLANHF is
set to zero.
A (input) 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 (workspace) REAL array, dimension (LWORK),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
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-trans‐
pose 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-trans‐
pose 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
LAPACK routine (version 3.2) November 2008 CLANHF(1)