1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
|
DOUBLE PRECISION FUNCTION ZLANHE( NORM, UPLO, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION WORK( * )
COMPLEX*16 A( LDA, * )
* ..
*
* Purpose
* =======
*
* ZLANHE 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.
*
* Description
* ===========
*
* ZLANHE returns the value
*
* ZLANHE = ( 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*1
* Specifies the value to be returned in ZLANHE as described
* above.
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* hermitian matrix A is to be referenced.
* = 'U': Upper triangular part of A is referenced
* = 'L': Lower triangular part of A is referenced
*
* N (input) INTEGER
* The order of the matrix A. N >= 0. When N = 0, ZLANHE is
* set to zero.
*
* A (input) COMPLEX*16 array, dimension (LDA,N)
* The hermitian matrix A. If UPLO = 'U', the leading n by n
* upper triangular part of A contains the upper triangular part
* of the matrix A, and the strictly lower triangular part of A
* is not referenced. If UPLO = 'L', the leading n by n lower
* triangular part of A contains the lower triangular part of
* the matrix A, and the strictly upper triangular part of A is
* not referenced. Note that the imaginary parts of the diagonal
* elements need not be set and are assumed to be zero.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(N,1).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (LWORK),
* where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
* WORK is not referenced.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL ZLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, J - 1
VALUE = MAX( VALUE, ABS( A( I, J ) ) )
10 CONTINUE
VALUE = MAX( VALUE, ABS( DBLE( A( J, J ) ) ) )
20 CONTINUE
ELSE
DO 40 J = 1, N
VALUE = MAX( VALUE, ABS( DBLE( A( J, J ) ) ) )
DO 30 I = J + 1, N
VALUE = MAX( VALUE, ABS( A( I, J ) ) )
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is hermitian).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
DO 50 I = 1, J - 1
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
50 CONTINUE
WORK( J ) = SUM + ABS( DBLE( A( J, J ) ) )
60 CONTINUE
DO 70 I = 1, N
VALUE = MAX( VALUE, WORK( I ) )
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( DBLE( A( J, J ) ) )
DO 90 I = J + 1, N
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
VALUE = MAX( VALUE, SUM )
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL ZLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL ZLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
120 CONTINUE
END IF
SUM = 2*SUM
DO 130 I = 1, N
IF( DBLE( A( I, I ) ).NE.ZERO ) THEN
ABSA = ABS( DBLE( A( I, I ) ) )
IF( SCALE.LT.ABSA ) THEN
SUM = ONE + SUM*( SCALE / ABSA )**2
SCALE = ABSA
ELSE
SUM = SUM + ( ABSA / SCALE )**2
END IF
END IF
130 CONTINUE
VALUE = SCALE*SQRT( SUM )
END IF
*
ZLANHE = VALUE
RETURN
*
* End of ZLANHE
*
END
|
