Annotation of rpl/lapack/lapack/dgeevx.f, revision 1.4
1.1 bertrand 1: SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
2: $ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
3: $ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
4: *
5: * -- LAPACK driver routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER BALANC, JOBVL, JOBVR, SENSE
12: INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
13: DOUBLE PRECISION ABNRM
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
18: $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
19: $ WI( * ), WORK( * ), WR( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
26: * eigenvalues and, optionally, the left and/or right eigenvectors.
27: *
28: * Optionally also, it computes a balancing transformation to improve
29: * the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
30: * SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
31: * (RCONDE), and reciprocal condition numbers for the right
32: * eigenvectors (RCONDV).
33: *
34: * The right eigenvector v(j) of A satisfies
35: * A * v(j) = lambda(j) * v(j)
36: * where lambda(j) is its eigenvalue.
37: * The left eigenvector u(j) of A satisfies
38: * u(j)**H * A = lambda(j) * u(j)**H
39: * where u(j)**H denotes the conjugate transpose of u(j).
40: *
41: * The computed eigenvectors are normalized to have Euclidean norm
42: * equal to 1 and largest component real.
43: *
44: * Balancing a matrix means permuting the rows and columns to make it
45: * more nearly upper triangular, and applying a diagonal similarity
46: * transformation D * A * D**(-1), where D is a diagonal matrix, to
47: * make its rows and columns closer in norm and the condition numbers
48: * of its eigenvalues and eigenvectors smaller. The computed
49: * reciprocal condition numbers correspond to the balanced matrix.
50: * Permuting rows and columns will not change the condition numbers
51: * (in exact arithmetic) but diagonal scaling will. For further
52: * explanation of balancing, see section 4.10.2 of the LAPACK
53: * Users' Guide.
54: *
55: * Arguments
56: * =========
57: *
58: * BALANC (input) CHARACTER*1
59: * Indicates how the input matrix should be diagonally scaled
60: * and/or permuted to improve the conditioning of its
61: * eigenvalues.
62: * = 'N': Do not diagonally scale or permute;
63: * = 'P': Perform permutations to make the matrix more nearly
64: * upper triangular. Do not diagonally scale;
65: * = 'S': Diagonally scale the matrix, i.e. replace A by
66: * D*A*D**(-1), where D is a diagonal matrix chosen
67: * to make the rows and columns of A more equal in
68: * norm. Do not permute;
69: * = 'B': Both diagonally scale and permute A.
70: *
71: * Computed reciprocal condition numbers will be for the matrix
72: * after balancing and/or permuting. Permuting does not change
73: * condition numbers (in exact arithmetic), but balancing does.
74: *
75: * JOBVL (input) CHARACTER*1
76: * = 'N': left eigenvectors of A are not computed;
77: * = 'V': left eigenvectors of A are computed.
78: * If SENSE = 'E' or 'B', JOBVL must = 'V'.
79: *
80: * JOBVR (input) CHARACTER*1
81: * = 'N': right eigenvectors of A are not computed;
82: * = 'V': right eigenvectors of A are computed.
83: * If SENSE = 'E' or 'B', JOBVR must = 'V'.
84: *
85: * SENSE (input) CHARACTER*1
86: * Determines which reciprocal condition numbers are computed.
87: * = 'N': None are computed;
88: * = 'E': Computed for eigenvalues only;
89: * = 'V': Computed for right eigenvectors only;
90: * = 'B': Computed for eigenvalues and right eigenvectors.
91: *
92: * If SENSE = 'E' or 'B', both left and right eigenvectors
93: * must also be computed (JOBVL = 'V' and JOBVR = 'V').
94: *
95: * N (input) INTEGER
96: * The order of the matrix A. N >= 0.
97: *
98: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
99: * On entry, the N-by-N matrix A.
100: * On exit, A has been overwritten. If JOBVL = 'V' or
101: * JOBVR = 'V', A contains the real Schur form of the balanced
102: * version of the input matrix A.
103: *
104: * LDA (input) INTEGER
105: * The leading dimension of the array A. LDA >= max(1,N).
106: *
107: * WR (output) DOUBLE PRECISION array, dimension (N)
108: * WI (output) DOUBLE PRECISION array, dimension (N)
109: * WR and WI contain the real and imaginary parts,
110: * respectively, of the computed eigenvalues. Complex
111: * conjugate pairs of eigenvalues will appear consecutively
112: * with the eigenvalue having the positive imaginary part
113: * first.
114: *
115: * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
116: * If JOBVL = 'V', the left eigenvectors u(j) are stored one
117: * after another in the columns of VL, in the same order
118: * as their eigenvalues.
119: * If JOBVL = 'N', VL is not referenced.
120: * If the j-th eigenvalue is real, then u(j) = VL(:,j),
121: * the j-th column of VL.
122: * If the j-th and (j+1)-st eigenvalues form a complex
123: * conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
124: * u(j+1) = VL(:,j) - i*VL(:,j+1).
125: *
126: * LDVL (input) INTEGER
127: * The leading dimension of the array VL. LDVL >= 1; if
128: * JOBVL = 'V', LDVL >= N.
129: *
130: * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
131: * If JOBVR = 'V', the right eigenvectors v(j) are stored one
132: * after another in the columns of VR, in the same order
133: * as their eigenvalues.
134: * If JOBVR = 'N', VR is not referenced.
135: * If the j-th eigenvalue is real, then v(j) = VR(:,j),
136: * the j-th column of VR.
137: * If the j-th and (j+1)-st eigenvalues form a complex
138: * conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
139: * v(j+1) = VR(:,j) - i*VR(:,j+1).
140: *
141: * LDVR (input) INTEGER
142: * The leading dimension of the array VR. LDVR >= 1, and if
143: * JOBVR = 'V', LDVR >= N.
144: *
145: * ILO (output) INTEGER
146: * IHI (output) INTEGER
147: * ILO and IHI are integer values determined when A was
148: * balanced. The balanced A(i,j) = 0 if I > J and
149: * J = 1,...,ILO-1 or I = IHI+1,...,N.
150: *
151: * SCALE (output) DOUBLE PRECISION array, dimension (N)
152: * Details of the permutations and scaling factors applied
153: * when balancing A. If P(j) is the index of the row and column
154: * interchanged with row and column j, and D(j) is the scaling
155: * factor applied to row and column j, then
156: * SCALE(J) = P(J), for J = 1,...,ILO-1
157: * = D(J), for J = ILO,...,IHI
158: * = P(J) for J = IHI+1,...,N.
159: * The order in which the interchanges are made is N to IHI+1,
160: * then 1 to ILO-1.
161: *
162: * ABNRM (output) DOUBLE PRECISION
163: * The one-norm of the balanced matrix (the maximum
164: * of the sum of absolute values of elements of any column).
165: *
166: * RCONDE (output) DOUBLE PRECISION array, dimension (N)
167: * RCONDE(j) is the reciprocal condition number of the j-th
168: * eigenvalue.
169: *
170: * RCONDV (output) DOUBLE PRECISION array, dimension (N)
171: * RCONDV(j) is the reciprocal condition number of the j-th
172: * right eigenvector.
173: *
174: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
175: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
176: *
177: * LWORK (input) INTEGER
178: * The dimension of the array WORK. If SENSE = 'N' or 'E',
179: * LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
180: * LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6).
181: * For good performance, LWORK must generally be larger.
182: *
183: * If LWORK = -1, then a workspace query is assumed; the routine
184: * only calculates the optimal size of the WORK array, returns
185: * this value as the first entry of the WORK array, and no error
186: * message related to LWORK is issued by XERBLA.
187: *
188: * IWORK (workspace) INTEGER array, dimension (2*N-2)
189: * If SENSE = 'N' or 'E', not referenced.
190: *
191: * INFO (output) INTEGER
192: * = 0: successful exit
193: * < 0: if INFO = -i, the i-th argument had an illegal value.
194: * > 0: if INFO = i, the QR algorithm failed to compute all the
195: * eigenvalues, and no eigenvectors or condition numbers
196: * have been computed; elements 1:ILO-1 and i+1:N of WR
197: * and WI contain eigenvalues which have converged.
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: DOUBLE PRECISION ZERO, ONE
203: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
204: * ..
205: * .. Local Scalars ..
206: LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
207: $ WNTSNN, WNTSNV
208: CHARACTER JOB, SIDE
209: INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
210: $ MINWRK, NOUT
211: DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
212: $ SN
213: * ..
214: * .. Local Arrays ..
215: LOGICAL SELECT( 1 )
216: DOUBLE PRECISION DUM( 1 )
217: * ..
218: * .. External Subroutines ..
219: EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
220: $ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
221: $ DTRSNA, XERBLA
222: * ..
223: * .. External Functions ..
224: LOGICAL LSAME
225: INTEGER IDAMAX, ILAENV
226: DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
227: EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
228: $ DNRM2
229: * ..
230: * .. Intrinsic Functions ..
231: INTRINSIC MAX, SQRT
232: * ..
233: * .. Executable Statements ..
234: *
235: * Test the input arguments
236: *
237: INFO = 0
238: LQUERY = ( LWORK.EQ.-1 )
239: WANTVL = LSAME( JOBVL, 'V' )
240: WANTVR = LSAME( JOBVR, 'V' )
241: WNTSNN = LSAME( SENSE, 'N' )
242: WNTSNE = LSAME( SENSE, 'E' )
243: WNTSNV = LSAME( SENSE, 'V' )
244: WNTSNB = LSAME( SENSE, 'B' )
245: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
246: $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
247: $ THEN
248: INFO = -1
249: ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
250: INFO = -2
251: ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
252: INFO = -3
253: ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
254: $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
255: $ WANTVR ) ) ) THEN
256: INFO = -4
257: ELSE IF( N.LT.0 ) THEN
258: INFO = -5
259: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260: INFO = -7
261: ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
262: INFO = -11
263: ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
264: INFO = -13
265: END IF
266: *
267: * Compute workspace
268: * (Note: Comments in the code beginning "Workspace:" describe the
269: * minimal amount of workspace needed at that point in the code,
270: * as well as the preferred amount for good performance.
271: * NB refers to the optimal block size for the immediately
272: * following subroutine, as returned by ILAENV.
273: * HSWORK refers to the workspace preferred by DHSEQR, as
274: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
275: * the worst case.)
276: *
277: IF( INFO.EQ.0 ) THEN
278: IF( N.EQ.0 ) THEN
279: MINWRK = 1
280: MAXWRK = 1
281: ELSE
282: MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
283: *
284: IF( WANTVL ) THEN
285: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
286: $ WORK, -1, INFO )
287: ELSE IF( WANTVR ) THEN
288: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
289: $ WORK, -1, INFO )
290: ELSE
291: IF( WNTSNN ) THEN
292: CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
293: $ LDVR, WORK, -1, INFO )
294: ELSE
295: CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
296: $ LDVR, WORK, -1, INFO )
297: END IF
298: END IF
299: HSWORK = WORK( 1 )
300: *
301: IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
302: MINWRK = 2*N
303: IF( .NOT.WNTSNN )
304: $ MINWRK = MAX( MINWRK, N*N+6*N )
305: MAXWRK = MAX( MAXWRK, HSWORK )
306: IF( .NOT.WNTSNN )
307: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
308: ELSE
309: MINWRK = 3*N
310: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
311: $ MINWRK = MAX( MINWRK, N*N + 6*N )
312: MAXWRK = MAX( MAXWRK, HSWORK )
313: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'DORGHR',
314: $ ' ', N, 1, N, -1 ) )
315: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
316: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
317: MAXWRK = MAX( MAXWRK, 3*N )
318: END IF
319: MAXWRK = MAX( MAXWRK, MINWRK )
320: END IF
321: WORK( 1 ) = MAXWRK
322: *
323: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
324: INFO = -21
325: END IF
326: END IF
327: *
328: IF( INFO.NE.0 ) THEN
329: CALL XERBLA( 'DGEEVX', -INFO )
330: RETURN
331: ELSE IF( LQUERY ) THEN
332: RETURN
333: END IF
334: *
335: * Quick return if possible
336: *
337: IF( N.EQ.0 )
338: $ RETURN
339: *
340: * Get machine constants
341: *
342: EPS = DLAMCH( 'P' )
343: SMLNUM = DLAMCH( 'S' )
344: BIGNUM = ONE / SMLNUM
345: CALL DLABAD( SMLNUM, BIGNUM )
346: SMLNUM = SQRT( SMLNUM ) / EPS
347: BIGNUM = ONE / SMLNUM
348: *
349: * Scale A if max element outside range [SMLNUM,BIGNUM]
350: *
351: ICOND = 0
352: ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
353: SCALEA = .FALSE.
354: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
355: SCALEA = .TRUE.
356: CSCALE = SMLNUM
357: ELSE IF( ANRM.GT.BIGNUM ) THEN
358: SCALEA = .TRUE.
359: CSCALE = BIGNUM
360: END IF
361: IF( SCALEA )
362: $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
363: *
364: * Balance the matrix and compute ABNRM
365: *
366: CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
367: ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
368: IF( SCALEA ) THEN
369: DUM( 1 ) = ABNRM
370: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
371: ABNRM = DUM( 1 )
372: END IF
373: *
374: * Reduce to upper Hessenberg form
375: * (Workspace: need 2*N, prefer N+N*NB)
376: *
377: ITAU = 1
378: IWRK = ITAU + N
379: CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
380: $ LWORK-IWRK+1, IERR )
381: *
382: IF( WANTVL ) THEN
383: *
384: * Want left eigenvectors
385: * Copy Householder vectors to VL
386: *
387: SIDE = 'L'
388: CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
389: *
390: * Generate orthogonal matrix in VL
391: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
392: *
393: CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
394: $ LWORK-IWRK+1, IERR )
395: *
396: * Perform QR iteration, accumulating Schur vectors in VL
397: * (Workspace: need 1, prefer HSWORK (see comments) )
398: *
399: IWRK = ITAU
400: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
401: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
402: *
403: IF( WANTVR ) THEN
404: *
405: * Want left and right eigenvectors
406: * Copy Schur vectors to VR
407: *
408: SIDE = 'B'
409: CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
410: END IF
411: *
412: ELSE IF( WANTVR ) THEN
413: *
414: * Want right eigenvectors
415: * Copy Householder vectors to VR
416: *
417: SIDE = 'R'
418: CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
419: *
420: * Generate orthogonal matrix in VR
421: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
422: *
423: CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
424: $ LWORK-IWRK+1, IERR )
425: *
426: * Perform QR iteration, accumulating Schur vectors in VR
427: * (Workspace: need 1, prefer HSWORK (see comments) )
428: *
429: IWRK = ITAU
430: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
431: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
432: *
433: ELSE
434: *
435: * Compute eigenvalues only
436: * If condition numbers desired, compute Schur form
437: *
438: IF( WNTSNN ) THEN
439: JOB = 'E'
440: ELSE
441: JOB = 'S'
442: END IF
443: *
444: * (Workspace: need 1, prefer HSWORK (see comments) )
445: *
446: IWRK = ITAU
447: CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
448: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
449: END IF
450: *
451: * If INFO > 0 from DHSEQR, then quit
452: *
453: IF( INFO.GT.0 )
454: $ GO TO 50
455: *
456: IF( WANTVL .OR. WANTVR ) THEN
457: *
458: * Compute left and/or right eigenvectors
459: * (Workspace: need 3*N)
460: *
461: CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
462: $ N, NOUT, WORK( IWRK ), IERR )
463: END IF
464: *
465: * Compute condition numbers if desired
466: * (Workspace: need N*N+6*N unless SENSE = 'E')
467: *
468: IF( .NOT.WNTSNN ) THEN
469: CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
470: $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
471: $ ICOND )
472: END IF
473: *
474: IF( WANTVL ) THEN
475: *
476: * Undo balancing of left eigenvectors
477: *
478: CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
479: $ IERR )
480: *
481: * Normalize left eigenvectors and make largest component real
482: *
483: DO 20 I = 1, N
484: IF( WI( I ).EQ.ZERO ) THEN
485: SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
486: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
487: ELSE IF( WI( I ).GT.ZERO ) THEN
488: SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
489: $ DNRM2( N, VL( 1, I+1 ), 1 ) )
490: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
491: CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
492: DO 10 K = 1, N
493: WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
494: 10 CONTINUE
495: K = IDAMAX( N, WORK, 1 )
496: CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
497: CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
498: VL( K, I+1 ) = ZERO
499: END IF
500: 20 CONTINUE
501: END IF
502: *
503: IF( WANTVR ) THEN
504: *
505: * Undo balancing of right eigenvectors
506: *
507: CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
508: $ IERR )
509: *
510: * Normalize right eigenvectors and make largest component real
511: *
512: DO 40 I = 1, N
513: IF( WI( I ).EQ.ZERO ) THEN
514: SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
515: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
516: ELSE IF( WI( I ).GT.ZERO ) THEN
517: SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
518: $ DNRM2( N, VR( 1, I+1 ), 1 ) )
519: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
520: CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
521: DO 30 K = 1, N
522: WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
523: 30 CONTINUE
524: K = IDAMAX( N, WORK, 1 )
525: CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
526: CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
527: VR( K, I+1 ) = ZERO
528: END IF
529: 40 CONTINUE
530: END IF
531: *
532: * Undo scaling if necessary
533: *
534: 50 CONTINUE
535: IF( SCALEA ) THEN
536: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
537: $ MAX( N-INFO, 1 ), IERR )
538: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
539: $ MAX( N-INFO, 1 ), IERR )
540: IF( INFO.EQ.0 ) THEN
541: IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
542: $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
543: $ IERR )
544: ELSE
545: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
546: $ IERR )
547: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
548: $ IERR )
549: END IF
550: END IF
551: *
552: WORK( 1 ) = MAXWRK
553: RETURN
554: *
555: * End of DGEEVX
556: *
557: END
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