1CLANHF(1)LAPACK routine (version 3.2) CLANHF(1)
2
6 CLANHF - returns the value of the one norm, or the Frobenius norm, or
7 the infinity norm, or the element of largest absolute value of a com‐
8 plex Hermitian matrix A in RFP format
9
11 REAL FUNCTION CLANHF( NORM, TRANSR, UPLO, N, A, WORK )
12 CHARACTER NORM, TRANSR, UPLO
14 INTEGER N
16 REAL WORK( 0: * )
18 COMPLEX A( 0: * )
20
22 CLANHF returns the value of the one norm, or the Frobenius norm, or
23 the infinity norm, or the element of largest absolute value of a
24 complex Hermitian matrix A in RFP format.
25
27 CLANHF returns the value
28 CLANHF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
29 (
30 ( norm1(A), NORM = '1', 'O' or 'o'
31 (
32 ( normI(A), NORM = 'I' or 'i'
33 (
34 ( normF(A), NORM = 'F', 'f', 'E' or 'e' where
35 norm1 denotes the one norm of a matrix (maximum column sum), normI
36 denotes the infinity norm of a matrix (maximum row sum) and normF
37 denotes the Frobenius norm of a matrix (square root of sum of
38 squares). Note that max(abs(A(i,j))) is not a matrix norm.
39
41 NORM (input) CHARACTER
42 Specifies the value to be returned in CLANHF as described
43 above.
44 TRANSR (input) CHARACTER
46 Specifies whether the RFP format of A is normal or conjugate-
47 transposed format. = 'N': RFP format is Normal
48 = 'C': RFP format is Conjugate-transposed
49 UPLO (input) CHARACTER
51 On entry, UPLO specifies whether the RFP matrix A came from
52 an upper or lower triangular matrix as follows: UPLO = 'U' or
53 'u' RFP A came from an upper triangular matrix UPLO = 'L' or
54 'l' RFP A came from a lower triangular matrix
55 N (input) INTEGER
57 The order of the matrix A. N >= 0. When N = 0, CLANHF is
58 set to zero.
59 A (input) COMPLEX*16 array, dimension ( N*(N+1)/2 );
61 On entry, the matrix A in RFP Format. RFP Format is described
62 by TRANSR, UPLO and N as follows:
63 If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
64 K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If TRANSR =
65 'C' then RFP is the Conjugate-transpose of RFP A as defined
66 when TRANSR = 'N'. The contents of RFP A are defined by UPLO
67 as follows: If UPLO = 'U' the RFP A contains the ( N*(N+1)/2 )
68 elements of upper packed A either in normal or conjugate-
69 transpose Format. If UPLO = 'L' the RFP A contains the (
70 N*(N+1) /2 ) elements of lower packed A either in normal or
71 conjugate-transpose Format. The LDA of RFP A is (N+1)/2 when
72 TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even
73 and is N when is odd. See the Note below for more details.
74 Unchanged on exit.
75 WORK (workspace) REAL array, dimension (LWORK),
77 where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
78 WORK is not referenced.
79
81 We first consider Standard Packed Format when N is even.
82 We give an example where N = 6.
83 AP is Upper AP is Lower
84 00 01 02 03 04 05 00
85 11 12 13 14 15 10 11
86 22 23 24 25 20 21 22
87 33 34 35 30 31 32 33
88 44 45 40 41 42 43 44
89 55 50 51 52 53 54 55
90 Let TRANSR = 'N'. RFP holds AP as follows:
91 For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
92 three columns of AP upper. The lower triangle A(4:6,0:2) consists of
93 conjugate-transpose of the first three columns of AP upper. For UPLO =
94 'L' the lower trapezoid A(1:6,0:2) consists of the first three columns
95 of AP lower. The upper triangle A(0:2,0:2) consists of conjugate-trans‐
96 pose of the last three columns of AP lower. To denote conjugate we
97 place -- above the element. This covers the case N even and TRANSR =
98 'N'.
99 RFP A RFP A
100 -- -- --
101 03 04 05 33 43 53
102 -- --
103 13 14 15 00 44 54
104 --
105 23 24 25 10 11 55
106 33 34 35 20 21 22
107 --
108 00 44 45 30 31 32
109 -- --
110 01 11 55 40 41 42
111 -- -- --
112 02 12 22 50 51 52
113 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
114 transpose of RFP A above. One therefore gets:
115 RFP A RFP A
116 -- -- -- -- -- -- -- -- -- --
117 03 13 23 33 00 01 02 33 00 10 20 30 40 50
118 -- -- -- -- -- -- -- -- -- --
119 04 14 24 34 44 11 12 43 44 11 21 31 41 51
120 -- -- -- -- -- -- -- -- -- --
121 05 15 25 35 45 55 22 53 54 55 22 32 42 52
122 We next consider Standard Packed Format when N is odd.
123 We give an example where N = 5.
124 AP is Upper AP is Lower
125 00 01 02 03 04 00
126 11 12 13 14 10 11
127 22 23 24 20 21 22
128 33 34 30 31 32 33
129 44 40 41 42 43 44
130 Let TRANSR = 'N'. RFP holds AP as follows:
131 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
132 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
133 conjugate-transpose of the first two columns of AP upper. For UPLO =
134 'L' the lower trapezoid A(0:4,0:2) consists of the first three columns
135 of AP lower. The upper triangle A(0:1,1:2) consists of conjugate-trans‐
136 pose of the last two columns of AP lower. To denote conjugate we
137 place -- above the element. This covers the case N odd and TRANSR =
138 'N'.
139 RFP A RFP A
140 -- --
141 02 03 04 00 33 43
142 --
143 12 13 14 10 11 44
144 22 23 24 20 21 22
145 --
146 00 33 34 30 31 32
147 -- --
148 01 11 44 40 41 42
149 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
150 transpose of RFP A above. One therefore gets:
151 RFP A RFP A
152 -- -- -- -- -- -- -- -- --
153 02 12 22 00 01 00 10 20 30 40 50
154 -- -- -- -- -- -- -- -- --
155 03 13 23 33 11 33 11 21 31 41 51
156 -- -- -- -- -- -- -- -- --
157 04 14 24 34 44 43 44 22 32 42 52
158 LAPACK routine (version 3.2) November 2008 CLANHF(1)