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