1: *> \brief \b ZGEBAL
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGEBAL + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebal.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER JOB
25: * INTEGER IHI, ILO, INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION SCALE( * )
29: * COMPLEX*16 A( LDA, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZGEBAL balances a general complex matrix A. This involves, first,
39: *> permuting A by a similarity transformation to isolate eigenvalues
40: *> in the first 1 to ILO-1 and last IHI+1 to N elements on the
41: *> diagonal; and second, applying a diagonal similarity transformation
42: *> to rows and columns ILO to IHI to make the rows and columns as
43: *> close in norm as possible. Both steps are optional.
44: *>
45: *> Balancing may reduce the 1-norm of the matrix, and improve the
46: *> accuracy of the computed eigenvalues and/or eigenvectors.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] JOB
53: *> \verbatim
54: *> JOB is CHARACTER*1
55: *> Specifies the operations to be performed on A:
56: *> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
57: *> for i = 1,...,N;
58: *> = 'P': permute only;
59: *> = 'S': scale only;
60: *> = 'B': both permute and scale.
61: *> \endverbatim
62: *>
63: *> \param[in] N
64: *> \verbatim
65: *> N is INTEGER
66: *> The order of the matrix A. N >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in,out] A
70: *> \verbatim
71: *> A is COMPLEX*16 array, dimension (LDA,N)
72: *> On entry, the input matrix A.
73: *> On exit, A is overwritten by the balanced matrix.
74: *> If JOB = 'N', A is not referenced.
75: *> See Further Details.
76: *> \endverbatim
77: *>
78: *> \param[in] LDA
79: *> \verbatim
80: *> LDA is INTEGER
81: *> The leading dimension of the array A. LDA >= max(1,N).
82: *> \endverbatim
83: *>
84: *> \param[out] ILO
85: *> \verbatim
86: *> \endverbatim
87: *>
88: *> \param[out] IHI
89: *> \verbatim
90: *> ILO and IHI are set to INTEGER such that on exit
91: *> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
92: *> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
93: *> \endverbatim
94: *>
95: *> \param[out] SCALE
96: *> \verbatim
97: *> SCALE is DOUBLE PRECISION array, dimension (N)
98: *> Details of the permutations and scaling factors applied to
99: *> A. If P(j) is the index of the row and column interchanged
100: *> with row and column j and D(j) is the scaling factor
101: *> applied to row and column j, then
102: *> SCALE(j) = P(j) for j = 1,...,ILO-1
103: *> = D(j) for j = ILO,...,IHI
104: *> = P(j) for j = IHI+1,...,N.
105: *> The order in which the interchanges are made is N to IHI+1,
106: *> then 1 to ILO-1.
107: *> \endverbatim
108: *>
109: *> \param[out] INFO
110: *> \verbatim
111: *> INFO is INTEGER
112: *> = 0: successful exit.
113: *> < 0: if INFO = -i, the i-th argument had an illegal value.
114: *> \endverbatim
115: *
116: * Authors:
117: * ========
118: *
119: *> \author Univ. of Tennessee
120: *> \author Univ. of California Berkeley
121: *> \author Univ. of Colorado Denver
122: *> \author NAG Ltd.
123: *
124: *> \date November 2015
125: *
126: *> \ingroup complex16GEcomputational
127: *
128: *> \par Further Details:
129: * =====================
130: *>
131: *> \verbatim
132: *>
133: *> The permutations consist of row and column interchanges which put
134: *> the matrix in the form
135: *>
136: *> ( T1 X Y )
137: *> P A P = ( 0 B Z )
138: *> ( 0 0 T2 )
139: *>
140: *> where T1 and T2 are upper triangular matrices whose eigenvalues lie
141: *> along the diagonal. The column indices ILO and IHI mark the starting
142: *> and ending columns of the submatrix B. Balancing consists of applying
143: *> a diagonal similarity transformation inv(D) * B * D to make the
144: *> 1-norms of each row of B and its corresponding column nearly equal.
145: *> The output matrix is
146: *>
147: *> ( T1 X*D Y )
148: *> ( 0 inv(D)*B*D inv(D)*Z ).
149: *> ( 0 0 T2 )
150: *>
151: *> Information about the permutations P and the diagonal matrix D is
152: *> returned in the vector SCALE.
153: *>
154: *> This subroutine is based on the EISPACK routine CBAL.
155: *>
156: *> Modified by Tzu-Yi Chen, Computer Science Division, University of
157: *> California at Berkeley, USA
158: *> \endverbatim
159: *>
160: * =====================================================================
161: SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
162: *
163: * -- LAPACK computational routine (version 3.6.0) --
164: * -- LAPACK is a software package provided by Univ. of Tennessee, --
165: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166: * November 2015
167: *
168: * .. Scalar Arguments ..
169: CHARACTER JOB
170: INTEGER IHI, ILO, INFO, LDA, N
171: * ..
172: * .. Array Arguments ..
173: DOUBLE PRECISION SCALE( * )
174: COMPLEX*16 A( LDA, * )
175: * ..
176: *
177: * =====================================================================
178: *
179: * .. Parameters ..
180: DOUBLE PRECISION ZERO, ONE
181: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
182: DOUBLE PRECISION SCLFAC
183: PARAMETER ( SCLFAC = 2.0D+0 )
184: DOUBLE PRECISION FACTOR
185: PARAMETER ( FACTOR = 0.95D+0 )
186: * ..
187: * .. Local Scalars ..
188: LOGICAL NOCONV
189: INTEGER I, ICA, IEXC, IRA, J, K, L, M
190: DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
191: $ SFMIN2
192: COMPLEX*16 CDUM
193: * ..
194: * .. External Functions ..
195: LOGICAL DISNAN, LSAME
196: INTEGER IZAMAX
197: DOUBLE PRECISION DLAMCH, DZNRM2
198: EXTERNAL DISNAN, LSAME, IZAMAX, DLAMCH, DZNRM2
199: * ..
200: * .. External Subroutines ..
201: EXTERNAL XERBLA, ZDSCAL, ZSWAP
202: * ..
203: * .. Intrinsic Functions ..
204: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
205: *
206: * Test the input parameters
207: *
208: INFO = 0
209: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
210: $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
211: INFO = -1
212: ELSE IF( N.LT.0 ) THEN
213: INFO = -2
214: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
215: INFO = -4
216: END IF
217: IF( INFO.NE.0 ) THEN
218: CALL XERBLA( 'ZGEBAL', -INFO )
219: RETURN
220: END IF
221: *
222: K = 1
223: L = N
224: *
225: IF( N.EQ.0 )
226: $ GO TO 210
227: *
228: IF( LSAME( JOB, 'N' ) ) THEN
229: DO 10 I = 1, N
230: SCALE( I ) = ONE
231: 10 CONTINUE
232: GO TO 210
233: END IF
234: *
235: IF( LSAME( JOB, 'S' ) )
236: $ GO TO 120
237: *
238: * Permutation to isolate eigenvalues if possible
239: *
240: GO TO 50
241: *
242: * Row and column exchange.
243: *
244: 20 CONTINUE
245: SCALE( M ) = J
246: IF( J.EQ.M )
247: $ GO TO 30
248: *
249: CALL ZSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
250: CALL ZSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
251: *
252: 30 CONTINUE
253: GO TO ( 40, 80 )IEXC
254: *
255: * Search for rows isolating an eigenvalue and push them down.
256: *
257: 40 CONTINUE
258: IF( L.EQ.1 )
259: $ GO TO 210
260: L = L - 1
261: *
262: 50 CONTINUE
263: DO 70 J = L, 1, -1
264: *
265: DO 60 I = 1, L
266: IF( I.EQ.J )
267: $ GO TO 60
268: IF( DBLE( A( J, I ) ).NE.ZERO .OR. DIMAG( A( J, I ) ).NE.
269: $ ZERO )GO TO 70
270: 60 CONTINUE
271: *
272: M = L
273: IEXC = 1
274: GO TO 20
275: 70 CONTINUE
276: *
277: GO TO 90
278: *
279: * Search for columns isolating an eigenvalue and push them left.
280: *
281: 80 CONTINUE
282: K = K + 1
283: *
284: 90 CONTINUE
285: DO 110 J = K, L
286: *
287: DO 100 I = K, L
288: IF( I.EQ.J )
289: $ GO TO 100
290: IF( DBLE( A( I, J ) ).NE.ZERO .OR. DIMAG( A( I, J ) ).NE.
291: $ ZERO )GO TO 110
292: 100 CONTINUE
293: *
294: M = K
295: IEXC = 2
296: GO TO 20
297: 110 CONTINUE
298: *
299: 120 CONTINUE
300: DO 130 I = K, L
301: SCALE( I ) = ONE
302: 130 CONTINUE
303: *
304: IF( LSAME( JOB, 'P' ) )
305: $ GO TO 210
306: *
307: * Balance the submatrix in rows K to L.
308: *
309: * Iterative loop for norm reduction
310: *
311: SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
312: SFMAX1 = ONE / SFMIN1
313: SFMIN2 = SFMIN1*SCLFAC
314: SFMAX2 = ONE / SFMIN2
315: 140 CONTINUE
316: NOCONV = .FALSE.
317: *
318: DO 200 I = K, L
319: *
320: C = DZNRM2( L-K+1, A( K, I ), 1 )
321: R = DZNRM2( L-K+1, A( I, K ), LDA )
322: ICA = IZAMAX( L, A( 1, I ), 1 )
323: CA = ABS( A( ICA, I ) )
324: IRA = IZAMAX( N-K+1, A( I, K ), LDA )
325: RA = ABS( A( I, IRA+K-1 ) )
326: *
327: * Guard against zero C or R due to underflow.
328: *
329: IF( C.EQ.ZERO .OR. R.EQ.ZERO )
330: $ GO TO 200
331: G = R / SCLFAC
332: F = ONE
333: S = C + R
334: 160 CONTINUE
335: IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
336: $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
337: IF( DISNAN( C+F+CA+R+G+RA ) ) THEN
338: *
339: * Exit if NaN to avoid infinite loop
340: *
341: INFO = -3
342: CALL XERBLA( 'ZGEBAL', -INFO )
343: RETURN
344: END IF
345: F = F*SCLFAC
346: C = C*SCLFAC
347: CA = CA*SCLFAC
348: R = R / SCLFAC
349: G = G / SCLFAC
350: RA = RA / SCLFAC
351: GO TO 160
352: *
353: 170 CONTINUE
354: G = C / SCLFAC
355: 180 CONTINUE
356: IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
357: $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
358: F = F / SCLFAC
359: C = C / SCLFAC
360: G = G / SCLFAC
361: CA = CA / SCLFAC
362: R = R*SCLFAC
363: RA = RA*SCLFAC
364: GO TO 180
365: *
366: * Now balance.
367: *
368: 190 CONTINUE
369: IF( ( C+R ).GE.FACTOR*S )
370: $ GO TO 200
371: IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
372: IF( F*SCALE( I ).LE.SFMIN1 )
373: $ GO TO 200
374: END IF
375: IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
376: IF( SCALE( I ).GE.SFMAX1 / F )
377: $ GO TO 200
378: END IF
379: G = ONE / F
380: SCALE( I ) = SCALE( I )*F
381: NOCONV = .TRUE.
382: *
383: CALL ZDSCAL( N-K+1, G, A( I, K ), LDA )
384: CALL ZDSCAL( L, F, A( 1, I ), 1 )
385: *
386: 200 CONTINUE
387: *
388: IF( NOCONV )
389: $ GO TO 140
390: *
391: 210 CONTINUE
392: ILO = K
393: IHI = L
394: *
395: RETURN
396: *
397: * End of ZGEBAL
398: *
399: END
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