1: *> \brief \b DLALN2
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLALN2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
22: * LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
23: *
24: * .. Scalar Arguments ..
25: * LOGICAL LTRANS
26: * INTEGER INFO, LDA, LDB, LDX, NA, NW
27: * DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLALN2 solves a system of the form (ca A - w D ) X = s B
40: *> or (ca A**T - w D) X = s B with possible scaling ("s") and
41: *> perturbation of A. (A**T means A-transpose.)
42: *>
43: *> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
44: *> real diagonal matrix, w is a real or complex value, and X and B are
45: *> NA x 1 matrices -- real if w is real, complex if w is complex. NA
46: *> may be 1 or 2.
47: *>
48: *> If w is complex, X and B are represented as NA x 2 matrices,
49: *> the first column of each being the real part and the second
50: *> being the imaginary part.
51: *>
52: *> "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
53: *> so chosen that X can be computed without overflow. X is further
54: *> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
55: *> than overflow.
56: *>
57: *> If both singular values of (ca A - w D) are less than SMIN,
58: *> SMIN*identity will be used instead of (ca A - w D). If only one
59: *> singular value is less than SMIN, one element of (ca A - w D) will be
60: *> perturbed enough to make the smallest singular value roughly SMIN.
61: *> If both singular values are at least SMIN, (ca A - w D) will not be
62: *> perturbed. In any case, the perturbation will be at most some small
63: *> multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
64: *> are computed by infinity-norm approximations, and thus will only be
65: *> correct to a factor of 2 or so.
66: *>
67: *> Note: all input quantities are assumed to be smaller than overflow
68: *> by a reasonable factor. (See BIGNUM.)
69: *> \endverbatim
70: *
71: * Arguments:
72: * ==========
73: *
74: *> \param[in] LTRANS
75: *> \verbatim
76: *> LTRANS is LOGICAL
77: *> =.TRUE.: A-transpose will be used.
78: *> =.FALSE.: A will be used (not transposed.)
79: *> \endverbatim
80: *>
81: *> \param[in] NA
82: *> \verbatim
83: *> NA is INTEGER
84: *> The size of the matrix A. It may (only) be 1 or 2.
85: *> \endverbatim
86: *>
87: *> \param[in] NW
88: *> \verbatim
89: *> NW is INTEGER
90: *> 1 if "w" is real, 2 if "w" is complex. It may only be 1
91: *> or 2.
92: *> \endverbatim
93: *>
94: *> \param[in] SMIN
95: *> \verbatim
96: *> SMIN is DOUBLE PRECISION
97: *> The desired lower bound on the singular values of A. This
98: *> should be a safe distance away from underflow or overflow,
99: *> say, between (underflow/machine precision) and (machine
100: *> precision * overflow ). (See BIGNUM and ULP.)
101: *> \endverbatim
102: *>
103: *> \param[in] CA
104: *> \verbatim
105: *> CA is DOUBLE PRECISION
106: *> The coefficient c, which A is multiplied by.
107: *> \endverbatim
108: *>
109: *> \param[in] A
110: *> \verbatim
111: *> A is DOUBLE PRECISION array, dimension (LDA,NA)
112: *> The NA x NA matrix A.
113: *> \endverbatim
114: *>
115: *> \param[in] LDA
116: *> \verbatim
117: *> LDA is INTEGER
118: *> The leading dimension of A. It must be at least NA.
119: *> \endverbatim
120: *>
121: *> \param[in] D1
122: *> \verbatim
123: *> D1 is DOUBLE PRECISION
124: *> The 1,1 element in the diagonal matrix D.
125: *> \endverbatim
126: *>
127: *> \param[in] D2
128: *> \verbatim
129: *> D2 is DOUBLE PRECISION
130: *> The 2,2 element in the diagonal matrix D. Not used if NW=1.
131: *> \endverbatim
132: *>
133: *> \param[in] B
134: *> \verbatim
135: *> B is DOUBLE PRECISION array, dimension (LDB,NW)
136: *> The NA x NW matrix B (right-hand side). If NW=2 ("w" is
137: *> complex), column 1 contains the real part of B and column 2
138: *> contains the imaginary part.
139: *> \endverbatim
140: *>
141: *> \param[in] LDB
142: *> \verbatim
143: *> LDB is INTEGER
144: *> The leading dimension of B. It must be at least NA.
145: *> \endverbatim
146: *>
147: *> \param[in] WR
148: *> \verbatim
149: *> WR is DOUBLE PRECISION
150: *> The real part of the scalar "w".
151: *> \endverbatim
152: *>
153: *> \param[in] WI
154: *> \verbatim
155: *> WI is DOUBLE PRECISION
156: *> The imaginary part of the scalar "w". Not used if NW=1.
157: *> \endverbatim
158: *>
159: *> \param[out] X
160: *> \verbatim
161: *> X is DOUBLE PRECISION array, dimension (LDX,NW)
162: *> The NA x NW matrix X (unknowns), as computed by DLALN2.
163: *> If NW=2 ("w" is complex), on exit, column 1 will contain
164: *> the real part of X and column 2 will contain the imaginary
165: *> part.
166: *> \endverbatim
167: *>
168: *> \param[in] LDX
169: *> \verbatim
170: *> LDX is INTEGER
171: *> The leading dimension of X. It must be at least NA.
172: *> \endverbatim
173: *>
174: *> \param[out] SCALE
175: *> \verbatim
176: *> SCALE is DOUBLE PRECISION
177: *> The scale factor that B must be multiplied by to insure
178: *> that overflow does not occur when computing X. Thus,
179: *> (ca A - w D) X will be SCALE*B, not B (ignoring
180: *> perturbations of A.) It will be at most 1.
181: *> \endverbatim
182: *>
183: *> \param[out] XNORM
184: *> \verbatim
185: *> XNORM is DOUBLE PRECISION
186: *> The infinity-norm of X, when X is regarded as an NA x NW
187: *> real matrix.
188: *> \endverbatim
189: *>
190: *> \param[out] INFO
191: *> \verbatim
192: *> INFO is INTEGER
193: *> An error flag. It will be set to zero if no error occurs,
194: *> a negative number if an argument is in error, or a positive
195: *> number if ca A - w D had to be perturbed.
196: *> The possible values are:
197: *> = 0: No error occurred, and (ca A - w D) did not have to be
198: *> perturbed.
199: *> = 1: (ca A - w D) had to be perturbed to make its smallest
200: *> (or only) singular value greater than SMIN.
201: *> NOTE: In the interests of speed, this routine does not
202: *> check the inputs for errors.
203: *> \endverbatim
204: *
205: * Authors:
206: * ========
207: *
208: *> \author Univ. of Tennessee
209: *> \author Univ. of California Berkeley
210: *> \author Univ. of Colorado Denver
211: *> \author NAG Ltd.
212: *
213: *> \date November 2011
214: *
215: *> \ingroup doubleOTHERauxiliary
216: *
217: * =====================================================================
218: SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
219: $ LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
220: *
221: * -- LAPACK auxiliary routine (version 3.4.0) --
222: * -- LAPACK is a software package provided by Univ. of Tennessee, --
223: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
224: * November 2011
225: *
226: * .. Scalar Arguments ..
227: LOGICAL LTRANS
228: INTEGER INFO, LDA, LDB, LDX, NA, NW
229: DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
230: * ..
231: * .. Array Arguments ..
232: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
233: * ..
234: *
235: * =====================================================================
236: *
237: * .. Parameters ..
238: DOUBLE PRECISION ZERO, ONE
239: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
240: DOUBLE PRECISION TWO
241: PARAMETER ( TWO = 2.0D0 )
242: * ..
243: * .. Local Scalars ..
244: INTEGER ICMAX, J
245: DOUBLE PRECISION BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
246: $ CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
247: $ LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
248: $ UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
249: $ UR22, XI1, XI2, XR1, XR2
250: * ..
251: * .. Local Arrays ..
252: LOGICAL RSWAP( 4 ), ZSWAP( 4 )
253: INTEGER IPIVOT( 4, 4 )
254: DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
255: * ..
256: * .. External Functions ..
257: DOUBLE PRECISION DLAMCH
258: EXTERNAL DLAMCH
259: * ..
260: * .. External Subroutines ..
261: EXTERNAL DLADIV
262: * ..
263: * .. Intrinsic Functions ..
264: INTRINSIC ABS, MAX
265: * ..
266: * .. Equivalences ..
267: EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ),
268: $ ( CR( 1, 1 ), CRV( 1 ) )
269: * ..
270: * .. Data statements ..
271: DATA ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
272: DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
273: DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
274: $ 3, 2, 1 /
275: * ..
276: * .. Executable Statements ..
277: *
278: * Compute BIGNUM
279: *
280: SMLNUM = TWO*DLAMCH( 'Safe minimum' )
281: BIGNUM = ONE / SMLNUM
282: SMINI = MAX( SMIN, SMLNUM )
283: *
284: * Don't check for input errors
285: *
286: INFO = 0
287: *
288: * Standard Initializations
289: *
290: SCALE = ONE
291: *
292: IF( NA.EQ.1 ) THEN
293: *
294: * 1 x 1 (i.e., scalar) system C X = B
295: *
296: IF( NW.EQ.1 ) THEN
297: *
298: * Real 1x1 system.
299: *
300: * C = ca A - w D
301: *
302: CSR = CA*A( 1, 1 ) - WR*D1
303: CNORM = ABS( CSR )
304: *
305: * If | C | < SMINI, use C = SMINI
306: *
307: IF( CNORM.LT.SMINI ) THEN
308: CSR = SMINI
309: CNORM = SMINI
310: INFO = 1
311: END IF
312: *
313: * Check scaling for X = B / C
314: *
315: BNORM = ABS( B( 1, 1 ) )
316: IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
317: IF( BNORM.GT.BIGNUM*CNORM )
318: $ SCALE = ONE / BNORM
319: END IF
320: *
321: * Compute X
322: *
323: X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
324: XNORM = ABS( X( 1, 1 ) )
325: ELSE
326: *
327: * Complex 1x1 system (w is complex)
328: *
329: * C = ca A - w D
330: *
331: CSR = CA*A( 1, 1 ) - WR*D1
332: CSI = -WI*D1
333: CNORM = ABS( CSR ) + ABS( CSI )
334: *
335: * If | C | < SMINI, use C = SMINI
336: *
337: IF( CNORM.LT.SMINI ) THEN
338: CSR = SMINI
339: CSI = ZERO
340: CNORM = SMINI
341: INFO = 1
342: END IF
343: *
344: * Check scaling for X = B / C
345: *
346: BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
347: IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
348: IF( BNORM.GT.BIGNUM*CNORM )
349: $ SCALE = ONE / BNORM
350: END IF
351: *
352: * Compute X
353: *
354: CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
355: $ X( 1, 1 ), X( 1, 2 ) )
356: XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
357: END IF
358: *
359: ELSE
360: *
361: * 2x2 System
362: *
363: * Compute the real part of C = ca A - w D (or ca A**T - w D )
364: *
365: CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
366: CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
367: IF( LTRANS ) THEN
368: CR( 1, 2 ) = CA*A( 2, 1 )
369: CR( 2, 1 ) = CA*A( 1, 2 )
370: ELSE
371: CR( 2, 1 ) = CA*A( 2, 1 )
372: CR( 1, 2 ) = CA*A( 1, 2 )
373: END IF
374: *
375: IF( NW.EQ.1 ) THEN
376: *
377: * Real 2x2 system (w is real)
378: *
379: * Find the largest element in C
380: *
381: CMAX = ZERO
382: ICMAX = 0
383: *
384: DO 10 J = 1, 4
385: IF( ABS( CRV( J ) ).GT.CMAX ) THEN
386: CMAX = ABS( CRV( J ) )
387: ICMAX = J
388: END IF
389: 10 CONTINUE
390: *
391: * If norm(C) < SMINI, use SMINI*identity.
392: *
393: IF( CMAX.LT.SMINI ) THEN
394: BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
395: IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
396: IF( BNORM.GT.BIGNUM*SMINI )
397: $ SCALE = ONE / BNORM
398: END IF
399: TEMP = SCALE / SMINI
400: X( 1, 1 ) = TEMP*B( 1, 1 )
401: X( 2, 1 ) = TEMP*B( 2, 1 )
402: XNORM = TEMP*BNORM
403: INFO = 1
404: RETURN
405: END IF
406: *
407: * Gaussian elimination with complete pivoting.
408: *
409: UR11 = CRV( ICMAX )
410: CR21 = CRV( IPIVOT( 2, ICMAX ) )
411: UR12 = CRV( IPIVOT( 3, ICMAX ) )
412: CR22 = CRV( IPIVOT( 4, ICMAX ) )
413: UR11R = ONE / UR11
414: LR21 = UR11R*CR21
415: UR22 = CR22 - UR12*LR21
416: *
417: * If smaller pivot < SMINI, use SMINI
418: *
419: IF( ABS( UR22 ).LT.SMINI ) THEN
420: UR22 = SMINI
421: INFO = 1
422: END IF
423: IF( RSWAP( ICMAX ) ) THEN
424: BR1 = B( 2, 1 )
425: BR2 = B( 1, 1 )
426: ELSE
427: BR1 = B( 1, 1 )
428: BR2 = B( 2, 1 )
429: END IF
430: BR2 = BR2 - LR21*BR1
431: BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
432: IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
433: IF( BBND.GE.BIGNUM*ABS( UR22 ) )
434: $ SCALE = ONE / BBND
435: END IF
436: *
437: XR2 = ( BR2*SCALE ) / UR22
438: XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
439: IF( ZSWAP( ICMAX ) ) THEN
440: X( 1, 1 ) = XR2
441: X( 2, 1 ) = XR1
442: ELSE
443: X( 1, 1 ) = XR1
444: X( 2, 1 ) = XR2
445: END IF
446: XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
447: *
448: * Further scaling if norm(A) norm(X) > overflow
449: *
450: IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
451: IF( XNORM.GT.BIGNUM / CMAX ) THEN
452: TEMP = CMAX / BIGNUM
453: X( 1, 1 ) = TEMP*X( 1, 1 )
454: X( 2, 1 ) = TEMP*X( 2, 1 )
455: XNORM = TEMP*XNORM
456: SCALE = TEMP*SCALE
457: END IF
458: END IF
459: ELSE
460: *
461: * Complex 2x2 system (w is complex)
462: *
463: * Find the largest element in C
464: *
465: CI( 1, 1 ) = -WI*D1
466: CI( 2, 1 ) = ZERO
467: CI( 1, 2 ) = ZERO
468: CI( 2, 2 ) = -WI*D2
469: CMAX = ZERO
470: ICMAX = 0
471: *
472: DO 20 J = 1, 4
473: IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
474: CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
475: ICMAX = J
476: END IF
477: 20 CONTINUE
478: *
479: * If norm(C) < SMINI, use SMINI*identity.
480: *
481: IF( CMAX.LT.SMINI ) THEN
482: BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
483: $ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
484: IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
485: IF( BNORM.GT.BIGNUM*SMINI )
486: $ SCALE = ONE / BNORM
487: END IF
488: TEMP = SCALE / SMINI
489: X( 1, 1 ) = TEMP*B( 1, 1 )
490: X( 2, 1 ) = TEMP*B( 2, 1 )
491: X( 1, 2 ) = TEMP*B( 1, 2 )
492: X( 2, 2 ) = TEMP*B( 2, 2 )
493: XNORM = TEMP*BNORM
494: INFO = 1
495: RETURN
496: END IF
497: *
498: * Gaussian elimination with complete pivoting.
499: *
500: UR11 = CRV( ICMAX )
501: UI11 = CIV( ICMAX )
502: CR21 = CRV( IPIVOT( 2, ICMAX ) )
503: CI21 = CIV( IPIVOT( 2, ICMAX ) )
504: UR12 = CRV( IPIVOT( 3, ICMAX ) )
505: UI12 = CIV( IPIVOT( 3, ICMAX ) )
506: CR22 = CRV( IPIVOT( 4, ICMAX ) )
507: CI22 = CIV( IPIVOT( 4, ICMAX ) )
508: IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
509: *
510: * Code when off-diagonals of pivoted C are real
511: *
512: IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
513: TEMP = UI11 / UR11
514: UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
515: UI11R = -TEMP*UR11R
516: ELSE
517: TEMP = UR11 / UI11
518: UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
519: UR11R = -TEMP*UI11R
520: END IF
521: LR21 = CR21*UR11R
522: LI21 = CR21*UI11R
523: UR12S = UR12*UR11R
524: UI12S = UR12*UI11R
525: UR22 = CR22 - UR12*LR21
526: UI22 = CI22 - UR12*LI21
527: ELSE
528: *
529: * Code when diagonals of pivoted C are real
530: *
531: UR11R = ONE / UR11
532: UI11R = ZERO
533: LR21 = CR21*UR11R
534: LI21 = CI21*UR11R
535: UR12S = UR12*UR11R
536: UI12S = UI12*UR11R
537: UR22 = CR22 - UR12*LR21 + UI12*LI21
538: UI22 = -UR12*LI21 - UI12*LR21
539: END IF
540: U22ABS = ABS( UR22 ) + ABS( UI22 )
541: *
542: * If smaller pivot < SMINI, use SMINI
543: *
544: IF( U22ABS.LT.SMINI ) THEN
545: UR22 = SMINI
546: UI22 = ZERO
547: INFO = 1
548: END IF
549: IF( RSWAP( ICMAX ) ) THEN
550: BR2 = B( 1, 1 )
551: BR1 = B( 2, 1 )
552: BI2 = B( 1, 2 )
553: BI1 = B( 2, 2 )
554: ELSE
555: BR1 = B( 1, 1 )
556: BR2 = B( 2, 1 )
557: BI1 = B( 1, 2 )
558: BI2 = B( 2, 2 )
559: END IF
560: BR2 = BR2 - LR21*BR1 + LI21*BI1
561: BI2 = BI2 - LI21*BR1 - LR21*BI1
562: BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
563: $ ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
564: $ ABS( BR2 )+ABS( BI2 ) )
565: IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
566: IF( BBND.GE.BIGNUM*U22ABS ) THEN
567: SCALE = ONE / BBND
568: BR1 = SCALE*BR1
569: BI1 = SCALE*BI1
570: BR2 = SCALE*BR2
571: BI2 = SCALE*BI2
572: END IF
573: END IF
574: *
575: CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
576: XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
577: XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
578: IF( ZSWAP( ICMAX ) ) THEN
579: X( 1, 1 ) = XR2
580: X( 2, 1 ) = XR1
581: X( 1, 2 ) = XI2
582: X( 2, 2 ) = XI1
583: ELSE
584: X( 1, 1 ) = XR1
585: X( 2, 1 ) = XR2
586: X( 1, 2 ) = XI1
587: X( 2, 2 ) = XI2
588: END IF
589: XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
590: *
591: * Further scaling if norm(A) norm(X) > overflow
592: *
593: IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
594: IF( XNORM.GT.BIGNUM / CMAX ) THEN
595: TEMP = CMAX / BIGNUM
596: X( 1, 1 ) = TEMP*X( 1, 1 )
597: X( 2, 1 ) = TEMP*X( 2, 1 )
598: X( 1, 2 ) = TEMP*X( 1, 2 )
599: X( 2, 2 ) = TEMP*X( 2, 2 )
600: XNORM = TEMP*XNORM
601: SCALE = TEMP*SCALE
602: END IF
603: END IF
604: END IF
605: END IF
606: *
607: RETURN
608: *
609: * End of DLALN2
610: *
611: END
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