1 DOUBLE PRECISION FUNCTION ZLANHB( NORM, UPLO, N, K, AB, LDAB,
2 $ WORK )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER NORM, UPLO
11 INTEGER K, LDAB, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION WORK( * )
15 COMPLEX*16 AB( LDAB, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLANHB returns the value of the one norm, or the Frobenius norm, or
22 * the infinity norm, or the element of largest absolute value of an
23 * n by n hermitian band matrix A, with k super-diagonals.
24 *
25 * Description
26 * ===========
27 *
28 * ZLANHB returns the value
29 *
30 * ZLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
31 * (
32 * ( norm1(A), NORM = '1', 'O' or 'o'
33 * (
34 * ( normI(A), NORM = 'I' or 'i'
35 * (
36 * ( normF(A), NORM = 'F', 'f', 'E' or 'e'
37 *
38 * where norm1 denotes the one norm of a matrix (maximum column sum),
39 * normI denotes the infinity norm of a matrix (maximum row sum) and
40 * normF denotes the Frobenius norm of a matrix (square root of sum of
41 * squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
42 *
43 * Arguments
44 * =========
45 *
46 * NORM (input) CHARACTER*1
47 * Specifies the value to be returned in ZLANHB as described
48 * above.
49 *
50 * UPLO (input) CHARACTER*1
51 * Specifies whether the upper or lower triangular part of the
52 * band matrix A is supplied.
53 * = 'U': Upper triangular
54 * = 'L': Lower triangular
55 *
56 * N (input) INTEGER
57 * The order of the matrix A. N >= 0. When N = 0, ZLANHB is
58 * set to zero.
59 *
60 * K (input) INTEGER
61 * The number of super-diagonals or sub-diagonals of the
62 * band matrix A. K >= 0.
63 *
64 * AB (input) COMPLEX*16 array, dimension (LDAB,N)
65 * The upper or lower triangle of the hermitian band matrix A,
66 * stored in the first K+1 rows of AB. The j-th column of A is
67 * stored in the j-th column of the array AB as follows:
68 * if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
69 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
70 * Note that the imaginary parts of the diagonal elements need
71 * not be set and are assumed to be zero.
72 *
73 * LDAB (input) INTEGER
74 * The leading dimension of the array AB. LDAB >= K+1.
75 *
76 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
77 * where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
78 * WORK is not referenced.
79 *
80 * =====================================================================
81 *
82 * .. Parameters ..
83 DOUBLE PRECISION ONE, ZERO
84 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
85 * ..
86 * .. Local Scalars ..
87 INTEGER I, J, L
88 DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
89 * ..
90 * .. External Functions ..
91 LOGICAL LSAME
92 EXTERNAL LSAME
93 * ..
94 * .. External Subroutines ..
95 EXTERNAL ZLASSQ
96 * ..
97 * .. Intrinsic Functions ..
98 INTRINSIC ABS, DBLE, MAX, MIN, SQRT
99 * ..
100 * .. Executable Statements ..
101 *
102 IF( N.EQ.0 ) THEN
103 VALUE = ZERO
104 ELSE IF( LSAME( NORM, 'M' ) ) THEN
105 *
106 * Find max(abs(A(i,j))).
107 *
108 VALUE = ZERO
109 IF( LSAME( UPLO, 'U' ) ) THEN
110 DO 20 J = 1, N
111 DO 10 I = MAX( K+2-J, 1 ), K
112 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
113 10 CONTINUE
114 VALUE = MAX( VALUE, ABS( DBLE( AB( K+1, J ) ) ) )
115 20 CONTINUE
116 ELSE
117 DO 40 J = 1, N
118 VALUE = MAX( VALUE, ABS( DBLE( AB( 1, J ) ) ) )
119 DO 30 I = 2, MIN( N+1-J, K+1 )
120 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
121 30 CONTINUE
122 40 CONTINUE
123 END IF
124 ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
125 $ ( NORM.EQ.'1' ) ) THEN
126 *
127 * Find normI(A) ( = norm1(A), since A is hermitian).
128 *
129 VALUE = ZERO
130 IF( LSAME( UPLO, 'U' ) ) THEN
131 DO 60 J = 1, N
132 SUM = ZERO
133 L = K + 1 - J
134 DO 50 I = MAX( 1, J-K ), J - 1
135 ABSA = ABS( AB( L+I, J ) )
136 SUM = SUM + ABSA
137 WORK( I ) = WORK( I ) + ABSA
138 50 CONTINUE
139 WORK( J ) = SUM + ABS( DBLE( AB( K+1, J ) ) )
140 60 CONTINUE
141 DO 70 I = 1, N
142 VALUE = MAX( VALUE, WORK( I ) )
143 70 CONTINUE
144 ELSE
145 DO 80 I = 1, N
146 WORK( I ) = ZERO
147 80 CONTINUE
148 DO 100 J = 1, N
149 SUM = WORK( J ) + ABS( DBLE( AB( 1, J ) ) )
150 L = 1 - J
151 DO 90 I = J + 1, MIN( N, J+K )
152 ABSA = ABS( AB( L+I, J ) )
153 SUM = SUM + ABSA
154 WORK( I ) = WORK( I ) + ABSA
155 90 CONTINUE
156 VALUE = MAX( VALUE, SUM )
157 100 CONTINUE
158 END IF
159 ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
160 *
161 * Find normF(A).
162 *
163 SCALE = ZERO
164 SUM = ONE
165 IF( K.GT.0 ) THEN
166 IF( LSAME( UPLO, 'U' ) ) THEN
167 DO 110 J = 2, N
168 CALL ZLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
169 $ 1, SCALE, SUM )
170 110 CONTINUE
171 L = K + 1
172 ELSE
173 DO 120 J = 1, N - 1
174 CALL ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
175 $ SUM )
176 120 CONTINUE
177 L = 1
178 END IF
179 SUM = 2*SUM
180 ELSE
181 L = 1
182 END IF
183 DO 130 J = 1, N
184 IF( DBLE( AB( L, J ) ).NE.ZERO ) THEN
185 ABSA = ABS( DBLE( AB( L, J ) ) )
186 IF( SCALE.LT.ABSA ) THEN
187 SUM = ONE + SUM*( SCALE / ABSA )**2
188 SCALE = ABSA
189 ELSE
190 SUM = SUM + ( ABSA / SCALE )**2
191 END IF
192 END IF
193 130 CONTINUE
194 VALUE = SCALE*SQRT( SUM )
195 END IF
196 *
197 ZLANHB = VALUE
198 RETURN
199 *
200 * End of ZLANHB
201 *
202 END
2 $ WORK )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER NORM, UPLO
11 INTEGER K, LDAB, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION WORK( * )
15 COMPLEX*16 AB( LDAB, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLANHB returns the value of the one norm, or the Frobenius norm, or
22 * the infinity norm, or the element of largest absolute value of an
23 * n by n hermitian band matrix A, with k super-diagonals.
24 *
25 * Description
26 * ===========
27 *
28 * ZLANHB returns the value
29 *
30 * ZLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
31 * (
32 * ( norm1(A), NORM = '1', 'O' or 'o'
33 * (
34 * ( normI(A), NORM = 'I' or 'i'
35 * (
36 * ( normF(A), NORM = 'F', 'f', 'E' or 'e'
37 *
38 * where norm1 denotes the one norm of a matrix (maximum column sum),
39 * normI denotes the infinity norm of a matrix (maximum row sum) and
40 * normF denotes the Frobenius norm of a matrix (square root of sum of
41 * squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
42 *
43 * Arguments
44 * =========
45 *
46 * NORM (input) CHARACTER*1
47 * Specifies the value to be returned in ZLANHB as described
48 * above.
49 *
50 * UPLO (input) CHARACTER*1
51 * Specifies whether the upper or lower triangular part of the
52 * band matrix A is supplied.
53 * = 'U': Upper triangular
54 * = 'L': Lower triangular
55 *
56 * N (input) INTEGER
57 * The order of the matrix A. N >= 0. When N = 0, ZLANHB is
58 * set to zero.
59 *
60 * K (input) INTEGER
61 * The number of super-diagonals or sub-diagonals of the
62 * band matrix A. K >= 0.
63 *
64 * AB (input) COMPLEX*16 array, dimension (LDAB,N)
65 * The upper or lower triangle of the hermitian band matrix A,
66 * stored in the first K+1 rows of AB. The j-th column of A is
67 * stored in the j-th column of the array AB as follows:
68 * if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
69 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
70 * Note that the imaginary parts of the diagonal elements need
71 * not be set and are assumed to be zero.
72 *
73 * LDAB (input) INTEGER
74 * The leading dimension of the array AB. LDAB >= K+1.
75 *
76 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
77 * where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
78 * WORK is not referenced.
79 *
80 * =====================================================================
81 *
82 * .. Parameters ..
83 DOUBLE PRECISION ONE, ZERO
84 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
85 * ..
86 * .. Local Scalars ..
87 INTEGER I, J, L
88 DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
89 * ..
90 * .. External Functions ..
91 LOGICAL LSAME
92 EXTERNAL LSAME
93 * ..
94 * .. External Subroutines ..
95 EXTERNAL ZLASSQ
96 * ..
97 * .. Intrinsic Functions ..
98 INTRINSIC ABS, DBLE, MAX, MIN, SQRT
99 * ..
100 * .. Executable Statements ..
101 *
102 IF( N.EQ.0 ) THEN
103 VALUE = ZERO
104 ELSE IF( LSAME( NORM, 'M' ) ) THEN
105 *
106 * Find max(abs(A(i,j))).
107 *
108 VALUE = ZERO
109 IF( LSAME( UPLO, 'U' ) ) THEN
110 DO 20 J = 1, N
111 DO 10 I = MAX( K+2-J, 1 ), K
112 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
113 10 CONTINUE
114 VALUE = MAX( VALUE, ABS( DBLE( AB( K+1, J ) ) ) )
115 20 CONTINUE
116 ELSE
117 DO 40 J = 1, N
118 VALUE = MAX( VALUE, ABS( DBLE( AB( 1, J ) ) ) )
119 DO 30 I = 2, MIN( N+1-J, K+1 )
120 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
121 30 CONTINUE
122 40 CONTINUE
123 END IF
124 ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
125 $ ( NORM.EQ.'1' ) ) THEN
126 *
127 * Find normI(A) ( = norm1(A), since A is hermitian).
128 *
129 VALUE = ZERO
130 IF( LSAME( UPLO, 'U' ) ) THEN
131 DO 60 J = 1, N
132 SUM = ZERO
133 L = K + 1 - J
134 DO 50 I = MAX( 1, J-K ), J - 1
135 ABSA = ABS( AB( L+I, J ) )
136 SUM = SUM + ABSA
137 WORK( I ) = WORK( I ) + ABSA
138 50 CONTINUE
139 WORK( J ) = SUM + ABS( DBLE( AB( K+1, J ) ) )
140 60 CONTINUE
141 DO 70 I = 1, N
142 VALUE = MAX( VALUE, WORK( I ) )
143 70 CONTINUE
144 ELSE
145 DO 80 I = 1, N
146 WORK( I ) = ZERO
147 80 CONTINUE
148 DO 100 J = 1, N
149 SUM = WORK( J ) + ABS( DBLE( AB( 1, J ) ) )
150 L = 1 - J
151 DO 90 I = J + 1, MIN( N, J+K )
152 ABSA = ABS( AB( L+I, J ) )
153 SUM = SUM + ABSA
154 WORK( I ) = WORK( I ) + ABSA
155 90 CONTINUE
156 VALUE = MAX( VALUE, SUM )
157 100 CONTINUE
158 END IF
159 ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
160 *
161 * Find normF(A).
162 *
163 SCALE = ZERO
164 SUM = ONE
165 IF( K.GT.0 ) THEN
166 IF( LSAME( UPLO, 'U' ) ) THEN
167 DO 110 J = 2, N
168 CALL ZLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
169 $ 1, SCALE, SUM )
170 110 CONTINUE
171 L = K + 1
172 ELSE
173 DO 120 J = 1, N - 1
174 CALL ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
175 $ SUM )
176 120 CONTINUE
177 L = 1
178 END IF
179 SUM = 2*SUM
180 ELSE
181 L = 1
182 END IF
183 DO 130 J = 1, N
184 IF( DBLE( AB( L, J ) ).NE.ZERO ) THEN
185 ABSA = ABS( DBLE( AB( L, J ) ) )
186 IF( SCALE.LT.ABSA ) THEN
187 SUM = ONE + SUM*( SCALE / ABSA )**2
188 SCALE = ABSA
189 ELSE
190 SUM = SUM + ( ABSA / SCALE )**2
191 END IF
192 END IF
193 130 CONTINUE
194 VALUE = SCALE*SQRT( SUM )
195 END IF
196 *
197 ZLANHB = VALUE
198 RETURN
199 *
200 * End of ZLANHB
201 *
202 END