1 DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
2 $ LDAB, 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 DIAG, NORM, UPLO
11 INTEGER K, LDAB, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION AB( LDAB, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLANTB returns the value of the one norm, or the Frobenius norm, or
21 * the infinity norm, or the element of largest absolute value of an
22 * n by n triangular band matrix A, with ( k + 1 ) diagonals.
23 *
24 * Description
25 * ===========
26 *
27 * DLANTB returns the value
28 *
29 * DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
30 * (
31 * ( norm1(A), NORM = '1', 'O' or 'o'
32 * (
33 * ( normI(A), NORM = 'I' or 'i'
34 * (
35 * ( normF(A), NORM = 'F', 'f', 'E' or 'e'
36 *
37 * where norm1 denotes the one norm of a matrix (maximum column sum),
38 * normI denotes the infinity norm of a matrix (maximum row sum) and
39 * normF denotes the Frobenius norm of a matrix (square root of sum of
40 * squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
41 *
42 * Arguments
43 * =========
44 *
45 * NORM (input) CHARACTER*1
46 * Specifies the value to be returned in DLANTB as described
47 * above.
48 *
49 * UPLO (input) CHARACTER*1
50 * Specifies whether the matrix A is upper or lower triangular.
51 * = 'U': Upper triangular
52 * = 'L': Lower triangular
53 *
54 * DIAG (input) CHARACTER*1
55 * Specifies whether or not the matrix A is unit triangular.
56 * = 'N': Non-unit triangular
57 * = 'U': Unit triangular
58 *
59 * N (input) INTEGER
60 * The order of the matrix A. N >= 0. When N = 0, DLANTB is
61 * set to zero.
62 *
63 * K (input) INTEGER
64 * The number of super-diagonals of the matrix A if UPLO = 'U',
65 * or the number of sub-diagonals of the matrix A if UPLO = 'L'.
66 * K >= 0.
67 *
68 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
69 * The upper or lower triangular band matrix A, stored in the
70 * first k+1 rows of AB. The j-th column of A is stored
71 * in the j-th column of the array AB as follows:
72 * if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
73 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
74 * Note that when DIAG = 'U', the elements of the array AB
75 * corresponding to the diagonal elements of the matrix A are
76 * not referenced, but are assumed to be one.
77 *
78 * LDAB (input) INTEGER
79 * The leading dimension of the array AB. LDAB >= K+1.
80 *
81 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
82 * where LWORK >= N when NORM = 'I'; otherwise, WORK is not
83 * referenced.
84 *
85 * =====================================================================
86 *
87 * .. Parameters ..
88 DOUBLE PRECISION ONE, ZERO
89 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
90 * ..
91 * .. Local Scalars ..
92 LOGICAL UDIAG
93 INTEGER I, J, L
94 DOUBLE PRECISION SCALE, SUM, VALUE
95 * ..
96 * .. External Subroutines ..
97 EXTERNAL DLASSQ
98 * ..
99 * .. External Functions ..
100 LOGICAL LSAME
101 EXTERNAL LSAME
102 * ..
103 * .. Intrinsic Functions ..
104 INTRINSIC ABS, MAX, MIN, SQRT
105 * ..
106 * .. Executable Statements ..
107 *
108 IF( N.EQ.0 ) THEN
109 VALUE = ZERO
110 ELSE IF( LSAME( NORM, 'M' ) ) THEN
111 *
112 * Find max(abs(A(i,j))).
113 *
114 IF( LSAME( DIAG, 'U' ) ) THEN
115 VALUE = ONE
116 IF( LSAME( UPLO, 'U' ) ) THEN
117 DO 20 J = 1, N
118 DO 10 I = MAX( K+2-J, 1 ), K
119 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
120 10 CONTINUE
121 20 CONTINUE
122 ELSE
123 DO 40 J = 1, N
124 DO 30 I = 2, MIN( N+1-J, K+1 )
125 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
126 30 CONTINUE
127 40 CONTINUE
128 END IF
129 ELSE
130 VALUE = ZERO
131 IF( LSAME( UPLO, 'U' ) ) THEN
132 DO 60 J = 1, N
133 DO 50 I = MAX( K+2-J, 1 ), K + 1
134 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
135 50 CONTINUE
136 60 CONTINUE
137 ELSE
138 DO 80 J = 1, N
139 DO 70 I = 1, MIN( N+1-J, K+1 )
140 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
141 70 CONTINUE
142 80 CONTINUE
143 END IF
144 END IF
145 ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
146 *
147 * Find norm1(A).
148 *
149 VALUE = ZERO
150 UDIAG = LSAME( DIAG, 'U' )
151 IF( LSAME( UPLO, 'U' ) ) THEN
152 DO 110 J = 1, N
153 IF( UDIAG ) THEN
154 SUM = ONE
155 DO 90 I = MAX( K+2-J, 1 ), K
156 SUM = SUM + ABS( AB( I, J ) )
157 90 CONTINUE
158 ELSE
159 SUM = ZERO
160 DO 100 I = MAX( K+2-J, 1 ), K + 1
161 SUM = SUM + ABS( AB( I, J ) )
162 100 CONTINUE
163 END IF
164 VALUE = MAX( VALUE, SUM )
165 110 CONTINUE
166 ELSE
167 DO 140 J = 1, N
168 IF( UDIAG ) THEN
169 SUM = ONE
170 DO 120 I = 2, MIN( N+1-J, K+1 )
171 SUM = SUM + ABS( AB( I, J ) )
172 120 CONTINUE
173 ELSE
174 SUM = ZERO
175 DO 130 I = 1, MIN( N+1-J, K+1 )
176 SUM = SUM + ABS( AB( I, J ) )
177 130 CONTINUE
178 END IF
179 VALUE = MAX( VALUE, SUM )
180 140 CONTINUE
181 END IF
182 ELSE IF( LSAME( NORM, 'I' ) ) THEN
183 *
184 * Find normI(A).
185 *
186 VALUE = ZERO
187 IF( LSAME( UPLO, 'U' ) ) THEN
188 IF( LSAME( DIAG, 'U' ) ) THEN
189 DO 150 I = 1, N
190 WORK( I ) = ONE
191 150 CONTINUE
192 DO 170 J = 1, N
193 L = K + 1 - J
194 DO 160 I = MAX( 1, J-K ), J - 1
195 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
196 160 CONTINUE
197 170 CONTINUE
198 ELSE
199 DO 180 I = 1, N
200 WORK( I ) = ZERO
201 180 CONTINUE
202 DO 200 J = 1, N
203 L = K + 1 - J
204 DO 190 I = MAX( 1, J-K ), J
205 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
206 190 CONTINUE
207 200 CONTINUE
208 END IF
209 ELSE
210 IF( LSAME( DIAG, 'U' ) ) THEN
211 DO 210 I = 1, N
212 WORK( I ) = ONE
213 210 CONTINUE
214 DO 230 J = 1, N
215 L = 1 - J
216 DO 220 I = J + 1, MIN( N, J+K )
217 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
218 220 CONTINUE
219 230 CONTINUE
220 ELSE
221 DO 240 I = 1, N
222 WORK( I ) = ZERO
223 240 CONTINUE
224 DO 260 J = 1, N
225 L = 1 - J
226 DO 250 I = J, MIN( N, J+K )
227 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
228 250 CONTINUE
229 260 CONTINUE
230 END IF
231 END IF
232 DO 270 I = 1, N
233 VALUE = MAX( VALUE, WORK( I ) )
234 270 CONTINUE
235 ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
236 *
237 * Find normF(A).
238 *
239 IF( LSAME( UPLO, 'U' ) ) THEN
240 IF( LSAME( DIAG, 'U' ) ) THEN
241 SCALE = ONE
242 SUM = N
243 IF( K.GT.0 ) THEN
244 DO 280 J = 2, N
245 CALL DLASSQ( MIN( J-1, K ),
246 $ AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
247 $ SUM )
248 280 CONTINUE
249 END IF
250 ELSE
251 SCALE = ZERO
252 SUM = ONE
253 DO 290 J = 1, N
254 CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ),
255 $ 1, SCALE, SUM )
256 290 CONTINUE
257 END IF
258 ELSE
259 IF( LSAME( DIAG, 'U' ) ) THEN
260 SCALE = ONE
261 SUM = N
262 IF( K.GT.0 ) THEN
263 DO 300 J = 1, N - 1
264 CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
265 $ SUM )
266 300 CONTINUE
267 END IF
268 ELSE
269 SCALE = ZERO
270 SUM = ONE
271 DO 310 J = 1, N
272 CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE,
273 $ SUM )
274 310 CONTINUE
275 END IF
276 END IF
277 VALUE = SCALE*SQRT( SUM )
278 END IF
279 *
280 DLANTB = VALUE
281 RETURN
282 *
283 * End of DLANTB
284 *
285 END
2 $ LDAB, 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 DIAG, NORM, UPLO
11 INTEGER K, LDAB, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION AB( LDAB, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DLANTB returns the value of the one norm, or the Frobenius norm, or
21 * the infinity norm, or the element of largest absolute value of an
22 * n by n triangular band matrix A, with ( k + 1 ) diagonals.
23 *
24 * Description
25 * ===========
26 *
27 * DLANTB returns the value
28 *
29 * DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
30 * (
31 * ( norm1(A), NORM = '1', 'O' or 'o'
32 * (
33 * ( normI(A), NORM = 'I' or 'i'
34 * (
35 * ( normF(A), NORM = 'F', 'f', 'E' or 'e'
36 *
37 * where norm1 denotes the one norm of a matrix (maximum column sum),
38 * normI denotes the infinity norm of a matrix (maximum row sum) and
39 * normF denotes the Frobenius norm of a matrix (square root of sum of
40 * squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
41 *
42 * Arguments
43 * =========
44 *
45 * NORM (input) CHARACTER*1
46 * Specifies the value to be returned in DLANTB as described
47 * above.
48 *
49 * UPLO (input) CHARACTER*1
50 * Specifies whether the matrix A is upper or lower triangular.
51 * = 'U': Upper triangular
52 * = 'L': Lower triangular
53 *
54 * DIAG (input) CHARACTER*1
55 * Specifies whether or not the matrix A is unit triangular.
56 * = 'N': Non-unit triangular
57 * = 'U': Unit triangular
58 *
59 * N (input) INTEGER
60 * The order of the matrix A. N >= 0. When N = 0, DLANTB is
61 * set to zero.
62 *
63 * K (input) INTEGER
64 * The number of super-diagonals of the matrix A if UPLO = 'U',
65 * or the number of sub-diagonals of the matrix A if UPLO = 'L'.
66 * K >= 0.
67 *
68 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
69 * The upper or lower triangular band matrix A, stored in the
70 * first k+1 rows of AB. The j-th column of A is stored
71 * in the j-th column of the array AB as follows:
72 * if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
73 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
74 * Note that when DIAG = 'U', the elements of the array AB
75 * corresponding to the diagonal elements of the matrix A are
76 * not referenced, but are assumed to be one.
77 *
78 * LDAB (input) INTEGER
79 * The leading dimension of the array AB. LDAB >= K+1.
80 *
81 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
82 * where LWORK >= N when NORM = 'I'; otherwise, WORK is not
83 * referenced.
84 *
85 * =====================================================================
86 *
87 * .. Parameters ..
88 DOUBLE PRECISION ONE, ZERO
89 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
90 * ..
91 * .. Local Scalars ..
92 LOGICAL UDIAG
93 INTEGER I, J, L
94 DOUBLE PRECISION SCALE, SUM, VALUE
95 * ..
96 * .. External Subroutines ..
97 EXTERNAL DLASSQ
98 * ..
99 * .. External Functions ..
100 LOGICAL LSAME
101 EXTERNAL LSAME
102 * ..
103 * .. Intrinsic Functions ..
104 INTRINSIC ABS, MAX, MIN, SQRT
105 * ..
106 * .. Executable Statements ..
107 *
108 IF( N.EQ.0 ) THEN
109 VALUE = ZERO
110 ELSE IF( LSAME( NORM, 'M' ) ) THEN
111 *
112 * Find max(abs(A(i,j))).
113 *
114 IF( LSAME( DIAG, 'U' ) ) THEN
115 VALUE = ONE
116 IF( LSAME( UPLO, 'U' ) ) THEN
117 DO 20 J = 1, N
118 DO 10 I = MAX( K+2-J, 1 ), K
119 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
120 10 CONTINUE
121 20 CONTINUE
122 ELSE
123 DO 40 J = 1, N
124 DO 30 I = 2, MIN( N+1-J, K+1 )
125 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
126 30 CONTINUE
127 40 CONTINUE
128 END IF
129 ELSE
130 VALUE = ZERO
131 IF( LSAME( UPLO, 'U' ) ) THEN
132 DO 60 J = 1, N
133 DO 50 I = MAX( K+2-J, 1 ), K + 1
134 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
135 50 CONTINUE
136 60 CONTINUE
137 ELSE
138 DO 80 J = 1, N
139 DO 70 I = 1, MIN( N+1-J, K+1 )
140 VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
141 70 CONTINUE
142 80 CONTINUE
143 END IF
144 END IF
145 ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
146 *
147 * Find norm1(A).
148 *
149 VALUE = ZERO
150 UDIAG = LSAME( DIAG, 'U' )
151 IF( LSAME( UPLO, 'U' ) ) THEN
152 DO 110 J = 1, N
153 IF( UDIAG ) THEN
154 SUM = ONE
155 DO 90 I = MAX( K+2-J, 1 ), K
156 SUM = SUM + ABS( AB( I, J ) )
157 90 CONTINUE
158 ELSE
159 SUM = ZERO
160 DO 100 I = MAX( K+2-J, 1 ), K + 1
161 SUM = SUM + ABS( AB( I, J ) )
162 100 CONTINUE
163 END IF
164 VALUE = MAX( VALUE, SUM )
165 110 CONTINUE
166 ELSE
167 DO 140 J = 1, N
168 IF( UDIAG ) THEN
169 SUM = ONE
170 DO 120 I = 2, MIN( N+1-J, K+1 )
171 SUM = SUM + ABS( AB( I, J ) )
172 120 CONTINUE
173 ELSE
174 SUM = ZERO
175 DO 130 I = 1, MIN( N+1-J, K+1 )
176 SUM = SUM + ABS( AB( I, J ) )
177 130 CONTINUE
178 END IF
179 VALUE = MAX( VALUE, SUM )
180 140 CONTINUE
181 END IF
182 ELSE IF( LSAME( NORM, 'I' ) ) THEN
183 *
184 * Find normI(A).
185 *
186 VALUE = ZERO
187 IF( LSAME( UPLO, 'U' ) ) THEN
188 IF( LSAME( DIAG, 'U' ) ) THEN
189 DO 150 I = 1, N
190 WORK( I ) = ONE
191 150 CONTINUE
192 DO 170 J = 1, N
193 L = K + 1 - J
194 DO 160 I = MAX( 1, J-K ), J - 1
195 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
196 160 CONTINUE
197 170 CONTINUE
198 ELSE
199 DO 180 I = 1, N
200 WORK( I ) = ZERO
201 180 CONTINUE
202 DO 200 J = 1, N
203 L = K + 1 - J
204 DO 190 I = MAX( 1, J-K ), J
205 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
206 190 CONTINUE
207 200 CONTINUE
208 END IF
209 ELSE
210 IF( LSAME( DIAG, 'U' ) ) THEN
211 DO 210 I = 1, N
212 WORK( I ) = ONE
213 210 CONTINUE
214 DO 230 J = 1, N
215 L = 1 - J
216 DO 220 I = J + 1, MIN( N, J+K )
217 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
218 220 CONTINUE
219 230 CONTINUE
220 ELSE
221 DO 240 I = 1, N
222 WORK( I ) = ZERO
223 240 CONTINUE
224 DO 260 J = 1, N
225 L = 1 - J
226 DO 250 I = J, MIN( N, J+K )
227 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
228 250 CONTINUE
229 260 CONTINUE
230 END IF
231 END IF
232 DO 270 I = 1, N
233 VALUE = MAX( VALUE, WORK( I ) )
234 270 CONTINUE
235 ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
236 *
237 * Find normF(A).
238 *
239 IF( LSAME( UPLO, 'U' ) ) THEN
240 IF( LSAME( DIAG, 'U' ) ) THEN
241 SCALE = ONE
242 SUM = N
243 IF( K.GT.0 ) THEN
244 DO 280 J = 2, N
245 CALL DLASSQ( MIN( J-1, K ),
246 $ AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
247 $ SUM )
248 280 CONTINUE
249 END IF
250 ELSE
251 SCALE = ZERO
252 SUM = ONE
253 DO 290 J = 1, N
254 CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ),
255 $ 1, SCALE, SUM )
256 290 CONTINUE
257 END IF
258 ELSE
259 IF( LSAME( DIAG, 'U' ) ) THEN
260 SCALE = ONE
261 SUM = N
262 IF( K.GT.0 ) THEN
263 DO 300 J = 1, N - 1
264 CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
265 $ SUM )
266 300 CONTINUE
267 END IF
268 ELSE
269 SCALE = ZERO
270 SUM = ONE
271 DO 310 J = 1, N
272 CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE,
273 $ SUM )
274 310 CONTINUE
275 END IF
276 END IF
277 VALUE = SCALE*SQRT( SUM )
278 END IF
279 *
280 DLANTB = VALUE
281 RETURN
282 *
283 * End of DLANTB
284 *
285 END