1DLANSF(1)LAPACK routine (version 3.2) DLANSF(1)
2
6 DLANSF - 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 real
8 symmetric matrix A in RFP format
9
11 DOUBLE PRECISION FUNCTION DLANSF( NORM, TRANSR, UPLO, N, A, WORK )
12 CHARACTER NORM, TRANSR, UPLO
14 INTEGER N
16 DOUBLE PRECISION A( 0: * ), WORK( 0: * )
18
20 DLANSF returns the value of the one norm, or the Frobenius norm, or the
21 infinity norm, or the element of largest absolute value of a real sym‐
22 metric matrix A in RFP format.
23
25 DLANSF returns the value
26 DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
27 (
28 ( norm1(A), NORM = '1', 'O' or 'o'
29 (
30 ( normI(A), NORM = 'I' or 'i'
31 (
32 ( normF(A), NORM = 'F', 'f', 'E' or 'e' where
33 norm1 denotes the one norm of a matrix (maximum column sum), normI
34 denotes the infinity norm of a matrix (maximum row sum) and normF
35 denotes the Frobenius norm of a matrix (square root of sum of
36 squares). Note that max(abs(A(i,j))) is not a matrix norm.
37
39 NORM (input) CHARACTER
40 Specifies the value to be returned in DLANSF as described
41 above.
42 TRANSR (input) CHARACTER
44 Specifies whether the RFP format of A is normal or transposed
45 format. = 'N': RFP format is Normal;
46 = 'T': RFP format is Transpose.
47 UPLO (input) CHARACTER
49 On entry, UPLO specifies whether the RFP matrix A came from an
50 upper or lower triangular matrix as follows:
51 = 'U': RFP A came from an upper triangular matrix;
52 = 'L': RFP A came from a lower triangular matrix.
53 N (input) INTEGER
55 The order of the matrix A. N >= 0. When N = 0, DLANSF is set to
56 zero.
57 A (input) DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
59 On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
60 part of the symmetric matrix A stored in RFP format. See the
61 "Notes" below for more details. Unchanged on exit.
62 WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
64 where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK
65 is not referenced.
66
68 We first consider Rectangular Full Packed (RFP) Format when N is even.
69 We give an example where N = 6.
70 AP is Upper AP is Lower
71 00 01 02 03 04 05 00
72 11 12 13 14 15 10 11
73 22 23 24 25 20 21 22
74 33 34 35 30 31 32 33
75 44 45 40 41 42 43 44
76 55 50 51 52 53 54 55
77 Let TRANSR = 'N'. RFP holds AP as follows:
78 For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
79 three columns of AP upper. The lower triangle A(4:6,0:2) consists of
80 the transpose of the first three columns of AP upper.
81 For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
82 three columns of AP lower. The upper triangle A(0:2,0:2) consists of
83 the transpose of the last three columns of AP lower.
84 This covers the case N even and TRANSR = 'N'.
85 RFP A RFP A
86 03 04 05 33 43 53
87 13 14 15 00 44 54
88 23 24 25 10 11 55
89 33 34 35 20 21 22
90 00 44 45 30 31 32
91 01 11 55 40 41 42
92 02 12 22 50 51 52
93 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
94 RFP A above. One therefore gets:
95 RFP A RFP A
96 03 13 23 33 00 01 02 33 00 10 20 30 40 50
97 04 14 24 34 44 11 12 43 44 11 21 31 41 51
98 05 15 25 35 45 55 22 53 54 55 22 32 42 52
99 We first consider Rectangular Full Packed (RFP) Format when N is odd.
100 We give an example where N = 5.
101 AP is Upper AP is Lower
102 00 01 02 03 04 00
103 11 12 13 14 10 11
104 22 23 24 20 21 22
105 33 34 30 31 32 33
106 44 40 41 42 43 44
107 Let TRANSR = 'N'. RFP holds AP as follows:
108 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
109 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
110 the transpose of the first two columns of AP upper.
111 For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
112 three columns of AP lower. The upper triangle A(0:1,1:2) consists of
113 the transpose of the last two columns of AP lower.
114 This covers the case N odd and TRANSR = 'N'.
115 RFP A RFP A
116 02 03 04 00 33 43
117 12 13 14 10 11 44
118 22 23 24 20 21 22
119 00 33 34 30 31 32
120 01 11 44 40 41 42
121 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
122 RFP A above. One therefore gets:
123 RFP A RFP A
124 02 12 22 00 01 00 10 20 30 40 50
125 03 13 23 33 11 33 11 21 31 41 51
126 04 14 24 34 44 43 44 22 32 42 52
127 Reference
128 =========
129 LAPACK routine (version 3.2) November 2008 DLANSF(1)