|
2329
|
1 SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO ) |
|
|
2 * |
|
7034
|
3 * -- LAPACK routine (version 3.1) -- |
|
|
4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
|
|
5 * November 2006 |
|
2329
|
6 * |
|
|
7 * .. Scalar Arguments .. |
|
|
8 CHARACTER JOB |
|
|
9 INTEGER IHI, ILO, INFO, LDA, N |
|
|
10 * .. |
|
|
11 * .. Array Arguments .. |
|
|
12 DOUBLE PRECISION A( LDA, * ), SCALE( * ) |
|
|
13 * .. |
|
|
14 * |
|
|
15 * Purpose |
|
|
16 * ======= |
|
|
17 * |
|
|
18 * DGEBAL balances a general real matrix A. This involves, first, |
|
|
19 * permuting A by a similarity transformation to isolate eigenvalues |
|
|
20 * in the first 1 to ILO-1 and last IHI+1 to N elements on the |
|
|
21 * diagonal; and second, applying a diagonal similarity transformation |
|
|
22 * to rows and columns ILO to IHI to make the rows and columns as |
|
|
23 * close in norm as possible. Both steps are optional. |
|
|
24 * |
|
|
25 * Balancing may reduce the 1-norm of the matrix, and improve the |
|
|
26 * accuracy of the computed eigenvalues and/or eigenvectors. |
|
|
27 * |
|
|
28 * Arguments |
|
|
29 * ========= |
|
|
30 * |
|
|
31 * JOB (input) CHARACTER*1 |
|
|
32 * Specifies the operations to be performed on A: |
|
|
33 * = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 |
|
|
34 * for i = 1,...,N; |
|
|
35 * = 'P': permute only; |
|
|
36 * = 'S': scale only; |
|
|
37 * = 'B': both permute and scale. |
|
|
38 * |
|
|
39 * N (input) INTEGER |
|
|
40 * The order of the matrix A. N >= 0. |
|
|
41 * |
|
|
42 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
|
|
43 * On entry, the input matrix A. |
|
|
44 * On exit, A is overwritten by the balanced matrix. |
|
|
45 * If JOB = 'N', A is not referenced. |
|
|
46 * See Further Details. |
|
|
47 * |
|
|
48 * LDA (input) INTEGER |
|
|
49 * The leading dimension of the array A. LDA >= max(1,N). |
|
|
50 * |
|
|
51 * ILO (output) INTEGER |
|
|
52 * IHI (output) INTEGER |
|
|
53 * ILO and IHI are set to integers such that on exit |
|
|
54 * A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. |
|
|
55 * If JOB = 'N' or 'S', ILO = 1 and IHI = N. |
|
|
56 * |
|
|
57 * SCALE (output) DOUBLE PRECISION array, dimension (N) |
|
|
58 * Details of the permutations and scaling factors applied to |
|
|
59 * A. If P(j) is the index of the row and column interchanged |
|
|
60 * with row and column j and D(j) is the scaling factor |
|
|
61 * applied to row and column j, then |
|
|
62 * SCALE(j) = P(j) for j = 1,...,ILO-1 |
|
|
63 * = D(j) for j = ILO,...,IHI |
|
|
64 * = P(j) for j = IHI+1,...,N. |
|
|
65 * The order in which the interchanges are made is N to IHI+1, |
|
|
66 * then 1 to ILO-1. |
|
|
67 * |
|
|
68 * INFO (output) INTEGER |
|
|
69 * = 0: successful exit. |
|
|
70 * < 0: if INFO = -i, the i-th argument had an illegal value. |
|
|
71 * |
|
|
72 * Further Details |
|
|
73 * =============== |
|
|
74 * |
|
|
75 * The permutations consist of row and column interchanges which put |
|
|
76 * the matrix in the form |
|
|
77 * |
|
|
78 * ( T1 X Y ) |
|
|
79 * P A P = ( 0 B Z ) |
|
|
80 * ( 0 0 T2 ) |
|
|
81 * |
|
|
82 * where T1 and T2 are upper triangular matrices whose eigenvalues lie |
|
|
83 * along the diagonal. The column indices ILO and IHI mark the starting |
|
|
84 * and ending columns of the submatrix B. Balancing consists of applying |
|
|
85 * a diagonal similarity transformation inv(D) * B * D to make the |
|
|
86 * 1-norms of each row of B and its corresponding column nearly equal. |
|
|
87 * The output matrix is |
|
|
88 * |
|
|
89 * ( T1 X*D Y ) |
|
|
90 * ( 0 inv(D)*B*D inv(D)*Z ). |
|
|
91 * ( 0 0 T2 ) |
|
|
92 * |
|
|
93 * Information about the permutations P and the diagonal matrix D is |
|
|
94 * returned in the vector SCALE. |
|
|
95 * |
|
|
96 * This subroutine is based on the EISPACK routine BALANC. |
|
|
97 * |
|
3333
|
98 * Modified by Tzu-Yi Chen, Computer Science Division, University of |
|
|
99 * California at Berkeley, USA |
|
|
100 * |
|
2329
|
101 * ===================================================================== |
|
|
102 * |
|
|
103 * .. Parameters .. |
|
|
104 DOUBLE PRECISION ZERO, ONE |
|
|
105 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
|
|
106 DOUBLE PRECISION SCLFAC |
|
7034
|
107 PARAMETER ( SCLFAC = 2.0D+0 ) |
|
2329
|
108 DOUBLE PRECISION FACTOR |
|
|
109 PARAMETER ( FACTOR = 0.95D+0 ) |
|
|
110 * .. |
|
|
111 * .. Local Scalars .. |
|
|
112 LOGICAL NOCONV |
|
|
113 INTEGER I, ICA, IEXC, IRA, J, K, L, M |
|
|
114 DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1, |
|
|
115 $ SFMIN2 |
|
|
116 * .. |
|
|
117 * .. External Functions .. |
|
|
118 LOGICAL LSAME |
|
|
119 INTEGER IDAMAX |
|
|
120 DOUBLE PRECISION DLAMCH |
|
|
121 EXTERNAL LSAME, IDAMAX, DLAMCH |
|
|
122 * .. |
|
|
123 * .. External Subroutines .. |
|
|
124 EXTERNAL DSCAL, DSWAP, XERBLA |
|
|
125 * .. |
|
|
126 * .. Intrinsic Functions .. |
|
|
127 INTRINSIC ABS, MAX, MIN |
|
|
128 * .. |
|
|
129 * .. Executable Statements .. |
|
|
130 * |
|
|
131 * Test the input parameters |
|
|
132 * |
|
|
133 INFO = 0 |
|
|
134 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. |
|
|
135 $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN |
|
|
136 INFO = -1 |
|
|
137 ELSE IF( N.LT.0 ) THEN |
|
|
138 INFO = -2 |
|
|
139 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
|
|
140 INFO = -4 |
|
|
141 END IF |
|
|
142 IF( INFO.NE.0 ) THEN |
|
|
143 CALL XERBLA( 'DGEBAL', -INFO ) |
|
|
144 RETURN |
|
|
145 END IF |
|
|
146 * |
|
|
147 K = 1 |
|
|
148 L = N |
|
|
149 * |
|
|
150 IF( N.EQ.0 ) |
|
|
151 $ GO TO 210 |
|
|
152 * |
|
|
153 IF( LSAME( JOB, 'N' ) ) THEN |
|
|
154 DO 10 I = 1, N |
|
|
155 SCALE( I ) = ONE |
|
|
156 10 CONTINUE |
|
|
157 GO TO 210 |
|
|
158 END IF |
|
|
159 * |
|
|
160 IF( LSAME( JOB, 'S' ) ) |
|
|
161 $ GO TO 120 |
|
|
162 * |
|
|
163 * Permutation to isolate eigenvalues if possible |
|
|
164 * |
|
|
165 GO TO 50 |
|
|
166 * |
|
|
167 * Row and column exchange. |
|
|
168 * |
|
|
169 20 CONTINUE |
|
|
170 SCALE( M ) = J |
|
|
171 IF( J.EQ.M ) |
|
|
172 $ GO TO 30 |
|
|
173 * |
|
|
174 CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) |
|
|
175 CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA ) |
|
|
176 * |
|
|
177 30 CONTINUE |
|
|
178 GO TO ( 40, 80 )IEXC |
|
|
179 * |
|
|
180 * Search for rows isolating an eigenvalue and push them down. |
|
|
181 * |
|
|
182 40 CONTINUE |
|
|
183 IF( L.EQ.1 ) |
|
|
184 $ GO TO 210 |
|
|
185 L = L - 1 |
|
|
186 * |
|
|
187 50 CONTINUE |
|
|
188 DO 70 J = L, 1, -1 |
|
|
189 * |
|
|
190 DO 60 I = 1, L |
|
|
191 IF( I.EQ.J ) |
|
|
192 $ GO TO 60 |
|
|
193 IF( A( J, I ).NE.ZERO ) |
|
|
194 $ GO TO 70 |
|
|
195 60 CONTINUE |
|
|
196 * |
|
|
197 M = L |
|
|
198 IEXC = 1 |
|
|
199 GO TO 20 |
|
|
200 70 CONTINUE |
|
|
201 * |
|
|
202 GO TO 90 |
|
|
203 * |
|
|
204 * Search for columns isolating an eigenvalue and push them left. |
|
|
205 * |
|
|
206 80 CONTINUE |
|
|
207 K = K + 1 |
|
|
208 * |
|
|
209 90 CONTINUE |
|
|
210 DO 110 J = K, L |
|
|
211 * |
|
|
212 DO 100 I = K, L |
|
|
213 IF( I.EQ.J ) |
|
|
214 $ GO TO 100 |
|
|
215 IF( A( I, J ).NE.ZERO ) |
|
|
216 $ GO TO 110 |
|
|
217 100 CONTINUE |
|
|
218 * |
|
|
219 M = K |
|
|
220 IEXC = 2 |
|
|
221 GO TO 20 |
|
|
222 110 CONTINUE |
|
|
223 * |
|
|
224 120 CONTINUE |
|
|
225 DO 130 I = K, L |
|
|
226 SCALE( I ) = ONE |
|
|
227 130 CONTINUE |
|
|
228 * |
|
|
229 IF( LSAME( JOB, 'P' ) ) |
|
|
230 $ GO TO 210 |
|
|
231 * |
|
|
232 * Balance the submatrix in rows K to L. |
|
|
233 * |
|
|
234 * Iterative loop for norm reduction |
|
|
235 * |
|
|
236 SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' ) |
|
|
237 SFMAX1 = ONE / SFMIN1 |
|
|
238 SFMIN2 = SFMIN1*SCLFAC |
|
|
239 SFMAX2 = ONE / SFMIN2 |
|
|
240 140 CONTINUE |
|
|
241 NOCONV = .FALSE. |
|
|
242 * |
|
|
243 DO 200 I = K, L |
|
|
244 C = ZERO |
|
|
245 R = ZERO |
|
|
246 * |
|
|
247 DO 150 J = K, L |
|
|
248 IF( J.EQ.I ) |
|
|
249 $ GO TO 150 |
|
|
250 C = C + ABS( A( J, I ) ) |
|
|
251 R = R + ABS( A( I, J ) ) |
|
|
252 150 CONTINUE |
|
|
253 ICA = IDAMAX( L, A( 1, I ), 1 ) |
|
|
254 CA = ABS( A( ICA, I ) ) |
|
|
255 IRA = IDAMAX( N-K+1, A( I, K ), LDA ) |
|
|
256 RA = ABS( A( I, IRA+K-1 ) ) |
|
|
257 * |
|
|
258 * Guard against zero C or R due to underflow. |
|
|
259 * |
|
|
260 IF( C.EQ.ZERO .OR. R.EQ.ZERO ) |
|
|
261 $ GO TO 200 |
|
|
262 G = R / SCLFAC |
|
|
263 F = ONE |
|
|
264 S = C + R |
|
|
265 160 CONTINUE |
|
|
266 IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR. |
|
|
267 $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170 |
|
|
268 F = F*SCLFAC |
|
|
269 C = C*SCLFAC |
|
|
270 CA = CA*SCLFAC |
|
|
271 R = R / SCLFAC |
|
|
272 G = G / SCLFAC |
|
|
273 RA = RA / SCLFAC |
|
|
274 GO TO 160 |
|
|
275 * |
|
|
276 170 CONTINUE |
|
|
277 G = C / SCLFAC |
|
|
278 180 CONTINUE |
|
|
279 IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR. |
|
|
280 $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190 |
|
|
281 F = F / SCLFAC |
|
|
282 C = C / SCLFAC |
|
|
283 G = G / SCLFAC |
|
|
284 CA = CA / SCLFAC |
|
|
285 R = R*SCLFAC |
|
|
286 RA = RA*SCLFAC |
|
|
287 GO TO 180 |
|
|
288 * |
|
|
289 * Now balance. |
|
|
290 * |
|
|
291 190 CONTINUE |
|
|
292 IF( ( C+R ).GE.FACTOR*S ) |
|
|
293 $ GO TO 200 |
|
|
294 IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN |
|
|
295 IF( F*SCALE( I ).LE.SFMIN1 ) |
|
|
296 $ GO TO 200 |
|
|
297 END IF |
|
|
298 IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN |
|
|
299 IF( SCALE( I ).GE.SFMAX1 / F ) |
|
|
300 $ GO TO 200 |
|
|
301 END IF |
|
|
302 G = ONE / F |
|
|
303 SCALE( I ) = SCALE( I )*F |
|
|
304 NOCONV = .TRUE. |
|
|
305 * |
|
|
306 CALL DSCAL( N-K+1, G, A( I, K ), LDA ) |
|
|
307 CALL DSCAL( L, F, A( 1, I ), 1 ) |
|
|
308 * |
|
|
309 200 CONTINUE |
|
|
310 * |
|
|
311 IF( NOCONV ) |
|
|
312 $ GO TO 140 |
|
|
313 * |
|
|
314 210 CONTINUE |
|
|
315 ILO = K |
|
|
316 IHI = L |
|
|
317 * |
|
|
318 RETURN |
|
|
319 * |
|
|
320 * End of DGEBAL |
|
|
321 * |
|
|
322 END |
