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1.1 bertrand 1: *> \brief \b ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHETF2_RK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetf2_rk.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetf2_rk.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetf2_rk.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * COMPLEX*16 A( LDA, * ), E ( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *> ZHETF2_RK computes the factorization of a complex Hermitian matrix A
38: *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
39: *>
40: *> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
41: *>
42: *> where U (or L) is unit upper (or lower) triangular matrix,
43: *> U**H (or L**H) is the conjugate of U (or L), P is a permutation
44: *> matrix, P**T is the transpose of P, and D is Hermitian and block
45: *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
46: *>
47: *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
48: *> For more information see Further Details section.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] UPLO
55: *> \verbatim
56: *> UPLO is CHARACTER*1
57: *> Specifies whether the upper or lower triangular part of the
58: *> Hermitian matrix A is stored:
59: *> = 'U': Upper triangular
60: *> = 'L': Lower triangular
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 Hermitian matrix A.
73: *> If UPLO = 'U': the leading N-by-N upper triangular part
74: *> of A contains the upper triangular part of the matrix A,
75: *> and the strictly lower triangular part of A is not
76: *> referenced.
77: *>
78: *> If UPLO = 'L': the leading N-by-N lower triangular part
79: *> of A contains the lower triangular part of the matrix A,
80: *> and the strictly upper triangular part of A is not
81: *> referenced.
82: *>
83: *> On exit, contains:
84: *> a) ONLY diagonal elements of the Hermitian block diagonal
85: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
86: *> (superdiagonal (or subdiagonal) elements of D
87: *> are stored on exit in array E), and
88: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
89: *> If UPLO = 'L': factor L in the subdiagonal part of A.
90: *> \endverbatim
91: *>
92: *> \param[in] LDA
93: *> \verbatim
94: *> LDA is INTEGER
95: *> The leading dimension of the array A. LDA >= max(1,N).
96: *> \endverbatim
97: *>
98: *> \param[out] E
99: *> \verbatim
100: *> E is COMPLEX*16 array, dimension (N)
101: *> On exit, contains the superdiagonal (or subdiagonal)
102: *> elements of the Hermitian block diagonal matrix D
103: *> with 1-by-1 or 2-by-2 diagonal blocks, where
104: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
105: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
106: *>
107: *> NOTE: For 1-by-1 diagonal block D(k), where
108: *> 1 <= k <= N, the element E(k) is set to 0 in both
109: *> UPLO = 'U' or UPLO = 'L' cases.
110: *> \endverbatim
111: *>
112: *> \param[out] IPIV
113: *> \verbatim
114: *> IPIV is INTEGER array, dimension (N)
115: *> IPIV describes the permutation matrix P in the factorization
116: *> of matrix A as follows. The absolute value of IPIV(k)
117: *> represents the index of row and column that were
118: *> interchanged with the k-th row and column. The value of UPLO
119: *> describes the order in which the interchanges were applied.
120: *> Also, the sign of IPIV represents the block structure of
121: *> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
122: *> diagonal blocks which correspond to 1 or 2 interchanges
123: *> at each factorization step. For more info see Further
124: *> Details section.
125: *>
126: *> If UPLO = 'U',
127: *> ( in factorization order, k decreases from N to 1 ):
128: *> a) A single positive entry IPIV(k) > 0 means:
129: *> D(k,k) is a 1-by-1 diagonal block.
130: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
131: *> interchanged in the matrix A(1:N,1:N);
132: *> If IPIV(k) = k, no interchange occurred.
133: *>
134: *> b) A pair of consecutive negative entries
135: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
136: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
137: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
138: *> 1) If -IPIV(k) != k, rows and columns
139: *> k and -IPIV(k) were interchanged
140: *> in the matrix A(1:N,1:N).
141: *> If -IPIV(k) = k, no interchange occurred.
142: *> 2) If -IPIV(k-1) != k-1, rows and columns
143: *> k-1 and -IPIV(k-1) were interchanged
144: *> in the matrix A(1:N,1:N).
145: *> If -IPIV(k-1) = k-1, no interchange occurred.
146: *>
147: *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
148: *>
149: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
150: *>
151: *> If UPLO = 'L',
152: *> ( in factorization order, k increases from 1 to N ):
153: *> a) A single positive entry IPIV(k) > 0 means:
154: *> D(k,k) is a 1-by-1 diagonal block.
155: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
156: *> interchanged in the matrix A(1:N,1:N).
157: *> If IPIV(k) = k, no interchange occurred.
158: *>
159: *> b) A pair of consecutive negative entries
160: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
161: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
162: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
163: *> 1) If -IPIV(k) != k, rows and columns
164: *> k and -IPIV(k) were interchanged
165: *> in the matrix A(1:N,1:N).
166: *> If -IPIV(k) = k, no interchange occurred.
167: *> 2) If -IPIV(k+1) != k+1, rows and columns
168: *> k-1 and -IPIV(k-1) were interchanged
169: *> in the matrix A(1:N,1:N).
170: *> If -IPIV(k+1) = k+1, no interchange occurred.
171: *>
172: *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
173: *>
174: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
175: *> \endverbatim
176: *>
177: *> \param[out] INFO
178: *> \verbatim
179: *> INFO is INTEGER
180: *> = 0: successful exit
181: *>
182: *> < 0: If INFO = -k, the k-th argument had an illegal value
183: *>
184: *> > 0: If INFO = k, the matrix A is singular, because:
185: *> If UPLO = 'U': column k in the upper
186: *> triangular part of A contains all zeros.
187: *> If UPLO = 'L': column k in the lower
188: *> triangular part of A contains all zeros.
189: *>
190: *> Therefore D(k,k) is exactly zero, and superdiagonal
191: *> elements of column k of U (or subdiagonal elements of
192: *> column k of L ) are all zeros. The factorization has
193: *> been completed, but the block diagonal matrix D is
194: *> exactly singular, and division by zero will occur if
195: *> it is used to solve a system of equations.
196: *>
197: *> NOTE: INFO only stores the first occurrence of
198: *> a singularity, any subsequent occurrence of singularity
199: *> is not stored in INFO even though the factorization
200: *> always completes.
201: *> \endverbatim
202: *
203: * Authors:
204: * ========
205: *
206: *> \author Univ. of Tennessee
207: *> \author Univ. of California Berkeley
208: *> \author Univ. of Colorado Denver
209: *> \author NAG Ltd.
210: *
211: *> \ingroup complex16HEcomputational
212: *
213: *> \par Further Details:
214: * =====================
215: *>
216: *> \verbatim
217: *> TODO: put further details
218: *> \endverbatim
219: *
220: *> \par Contributors:
221: * ==================
222: *>
223: *> \verbatim
224: *>
225: *> December 2016, Igor Kozachenko,
226: *> Computer Science Division,
227: *> University of California, Berkeley
228: *>
229: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
230: *> School of Mathematics,
231: *> University of Manchester
232: *>
233: *> 01-01-96 - Based on modifications by
234: *> J. Lewis, Boeing Computer Services Company
235: *> A. Petitet, Computer Science Dept.,
236: *> Univ. of Tenn., Knoxville abd , USA
237: *> \endverbatim
238: *
239: * =====================================================================
240: SUBROUTINE ZHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
241: *
1.5 ! bertrand 242: * -- LAPACK computational routine --
1.1 bertrand 243: * -- LAPACK is a software package provided by Univ. of Tennessee, --
244: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
245: *
246: * .. Scalar Arguments ..
247: CHARACTER UPLO
248: INTEGER INFO, LDA, N
249: * ..
250: * .. Array Arguments ..
251: INTEGER IPIV( * )
252: COMPLEX*16 A( LDA, * ), E( * )
253: * ..
254: *
255: * ======================================================================
256: *
257: * .. Parameters ..
258: DOUBLE PRECISION ZERO, ONE
259: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
260: DOUBLE PRECISION EIGHT, SEVTEN
261: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
262: COMPLEX*16 CZERO
263: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) )
264: * ..
265: * .. Local Scalars ..
266: LOGICAL DONE, UPPER
267: INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP,
268: $ P
269: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, DTEMP,
270: $ ROWMAX, TT, SFMIN
271: COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, Z
272: * ..
273: * .. External Functions ..
274: *
275: LOGICAL LSAME
276: INTEGER IZAMAX
277: DOUBLE PRECISION DLAMCH, DLAPY2
278: EXTERNAL LSAME, IZAMAX, DLAMCH, DLAPY2
279: * ..
280: * .. External Subroutines ..
281: EXTERNAL XERBLA, ZDSCAL, ZHER, ZSWAP
282: * ..
283: * .. Intrinsic Functions ..
284: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT
285: * ..
286: * .. Statement Functions ..
287: DOUBLE PRECISION CABS1
288: * ..
289: * .. Statement Function definitions ..
290: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
291: * ..
292: * .. Executable Statements ..
293: *
294: * Test the input parameters.
295: *
296: INFO = 0
297: UPPER = LSAME( UPLO, 'U' )
298: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
299: INFO = -1
300: ELSE IF( N.LT.0 ) THEN
301: INFO = -2
302: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
303: INFO = -4
304: END IF
305: IF( INFO.NE.0 ) THEN
306: CALL XERBLA( 'ZHETF2_RK', -INFO )
307: RETURN
308: END IF
309: *
310: * Initialize ALPHA for use in choosing pivot block size.
311: *
312: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
313: *
314: * Compute machine safe minimum
315: *
316: SFMIN = DLAMCH( 'S' )
317: *
318: IF( UPPER ) THEN
319: *
320: * Factorize A as U*D*U**H using the upper triangle of A
321: *
1.4 bertrand 322: * Initialize the first entry of array E, where superdiagonal
1.1 bertrand 323: * elements of D are stored
324: *
325: E( 1 ) = CZERO
326: *
327: * K is the main loop index, decreasing from N to 1 in steps of
328: * 1 or 2
329: *
330: K = N
331: 10 CONTINUE
332: *
333: * If K < 1, exit from loop
334: *
335: IF( K.LT.1 )
336: $ GO TO 34
337: KSTEP = 1
338: P = K
339: *
340: * Determine rows and columns to be interchanged and whether
341: * a 1-by-1 or 2-by-2 pivot block will be used
342: *
343: ABSAKK = ABS( DBLE( A( K, K ) ) )
344: *
345: * IMAX is the row-index of the largest off-diagonal element in
346: * column K, and COLMAX is its absolute value.
347: * Determine both COLMAX and IMAX.
348: *
349: IF( K.GT.1 ) THEN
350: IMAX = IZAMAX( K-1, A( 1, K ), 1 )
351: COLMAX = CABS1( A( IMAX, K ) )
352: ELSE
353: COLMAX = ZERO
354: END IF
355: *
356: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
357: *
358: * Column K is zero or underflow: set INFO and continue
359: *
360: IF( INFO.EQ.0 )
361: $ INFO = K
362: KP = K
363: A( K, K ) = DBLE( A( K, K ) )
364: *
365: * Set E( K ) to zero
366: *
367: IF( K.GT.1 )
368: $ E( K ) = CZERO
369: *
370: ELSE
371: *
372: * ============================================================
373: *
374: * BEGIN pivot search
375: *
376: * Case(1)
377: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
378: * (used to handle NaN and Inf)
379: *
380: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
381: *
382: * no interchange, use 1-by-1 pivot block
383: *
384: KP = K
385: *
386: ELSE
387: *
388: DONE = .FALSE.
389: *
390: * Loop until pivot found
391: *
392: 12 CONTINUE
393: *
394: * BEGIN pivot search loop body
395: *
396: *
397: * JMAX is the column-index of the largest off-diagonal
398: * element in row IMAX, and ROWMAX is its absolute value.
399: * Determine both ROWMAX and JMAX.
400: *
401: IF( IMAX.NE.K ) THEN
402: JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ),
403: $ LDA )
404: ROWMAX = CABS1( A( IMAX, JMAX ) )
405: ELSE
406: ROWMAX = ZERO
407: END IF
408: *
409: IF( IMAX.GT.1 ) THEN
410: ITEMP = IZAMAX( IMAX-1, A( 1, IMAX ), 1 )
411: DTEMP = CABS1( A( ITEMP, IMAX ) )
412: IF( DTEMP.GT.ROWMAX ) THEN
413: ROWMAX = DTEMP
414: JMAX = ITEMP
415: END IF
416: END IF
417: *
418: * Case(2)
419: * Equivalent to testing for
1.5 ! bertrand 420: * ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
1.1 bertrand 421: * (used to handle NaN and Inf)
422: *
423: IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) )
424: $ .LT.ALPHA*ROWMAX ) ) THEN
425: *
426: * interchange rows and columns K and IMAX,
427: * use 1-by-1 pivot block
428: *
429: KP = IMAX
430: DONE = .TRUE.
431: *
432: * Case(3)
433: * Equivalent to testing for ROWMAX.EQ.COLMAX,
434: * (used to handle NaN and Inf)
435: *
436: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
437: $ THEN
438: *
439: * interchange rows and columns K-1 and IMAX,
440: * use 2-by-2 pivot block
441: *
442: KP = IMAX
443: KSTEP = 2
444: DONE = .TRUE.
445: *
446: * Case(4)
447: ELSE
448: *
449: * Pivot not found: set params and repeat
450: *
451: P = IMAX
452: COLMAX = ROWMAX
453: IMAX = JMAX
454: END IF
455: *
456: * END pivot search loop body
457: *
458: IF( .NOT.DONE ) GOTO 12
459: *
460: END IF
461: *
462: * END pivot search
463: *
464: * ============================================================
465: *
466: * KK is the column of A where pivoting step stopped
467: *
468: KK = K - KSTEP + 1
469: *
470: * For only a 2x2 pivot, interchange rows and columns K and P
471: * in the leading submatrix A(1:k,1:k)
472: *
473: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
474: * (1) Swap columnar parts
475: IF( P.GT.1 )
476: $ CALL ZSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
477: * (2) Swap and conjugate middle parts
478: DO 14 J = P + 1, K - 1
479: T = DCONJG( A( J, K ) )
480: A( J, K ) = DCONJG( A( P, J ) )
481: A( P, J ) = T
482: 14 CONTINUE
483: * (3) Swap and conjugate corner elements at row-col interserction
484: A( P, K ) = DCONJG( A( P, K ) )
485: * (4) Swap diagonal elements at row-col intersection
486: R1 = DBLE( A( K, K ) )
487: A( K, K ) = DBLE( A( P, P ) )
488: A( P, P ) = R1
489: *
490: * Convert upper triangle of A into U form by applying
491: * the interchanges in columns k+1:N.
492: *
493: IF( K.LT.N )
494: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
495: *
496: END IF
497: *
498: * For both 1x1 and 2x2 pivots, interchange rows and
499: * columns KK and KP in the leading submatrix A(1:k,1:k)
500: *
501: IF( KP.NE.KK ) THEN
502: * (1) Swap columnar parts
503: IF( KP.GT.1 )
504: $ CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
505: * (2) Swap and conjugate middle parts
506: DO 15 J = KP + 1, KK - 1
507: T = DCONJG( A( J, KK ) )
508: A( J, KK ) = DCONJG( A( KP, J ) )
509: A( KP, J ) = T
510: 15 CONTINUE
511: * (3) Swap and conjugate corner elements at row-col interserction
512: A( KP, KK ) = DCONJG( A( KP, KK ) )
513: * (4) Swap diagonal elements at row-col intersection
514: R1 = DBLE( A( KK, KK ) )
515: A( KK, KK ) = DBLE( A( KP, KP ) )
516: A( KP, KP ) = R1
517: *
518: IF( KSTEP.EQ.2 ) THEN
519: * (*) Make sure that diagonal element of pivot is real
520: A( K, K ) = DBLE( A( K, K ) )
521: * (5) Swap row elements
522: T = A( K-1, K )
523: A( K-1, K ) = A( KP, K )
524: A( KP, K ) = T
525: END IF
526: *
527: * Convert upper triangle of A into U form by applying
528: * the interchanges in columns k+1:N.
529: *
530: IF( K.LT.N )
531: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
532: $ LDA )
533: *
534: ELSE
535: * (*) Make sure that diagonal element of pivot is real
536: A( K, K ) = DBLE( A( K, K ) )
537: IF( KSTEP.EQ.2 )
538: $ A( K-1, K-1 ) = DBLE( A( K-1, K-1 ) )
539: END IF
540: *
541: * Update the leading submatrix
542: *
543: IF( KSTEP.EQ.1 ) THEN
544: *
545: * 1-by-1 pivot block D(k): column k now holds
546: *
547: * W(k) = U(k)*D(k)
548: *
549: * where U(k) is the k-th column of U
550: *
551: IF( K.GT.1 ) THEN
552: *
553: * Perform a rank-1 update of A(1:k-1,1:k-1) and
554: * store U(k) in column k
555: *
556: IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN
557: *
558: * Perform a rank-1 update of A(1:k-1,1:k-1) as
559: * A := A - U(k)*D(k)*U(k)**T
560: * = A - W(k)*1/D(k)*W(k)**T
561: *
562: D11 = ONE / DBLE( A( K, K ) )
563: CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
564: *
565: * Store U(k) in column k
566: *
567: CALL ZDSCAL( K-1, D11, A( 1, K ), 1 )
568: ELSE
569: *
570: * Store L(k) in column K
571: *
572: D11 = DBLE( A( K, K ) )
573: DO 16 II = 1, K - 1
574: A( II, K ) = A( II, K ) / D11
575: 16 CONTINUE
576: *
577: * Perform a rank-1 update of A(k+1:n,k+1:n) as
578: * A := A - U(k)*D(k)*U(k)**T
579: * = A - W(k)*(1/D(k))*W(k)**T
580: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
581: *
582: CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
583: END IF
584: *
585: * Store the superdiagonal element of D in array E
586: *
587: E( K ) = CZERO
588: *
589: END IF
590: *
591: ELSE
592: *
593: * 2-by-2 pivot block D(k): columns k and k-1 now hold
594: *
595: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
596: *
597: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
598: * of U
599: *
600: * Perform a rank-2 update of A(1:k-2,1:k-2) as
601: *
602: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
603: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
604: *
605: * and store L(k) and L(k+1) in columns k and k+1
606: *
607: IF( K.GT.2 ) THEN
608: * D = |A12|
609: D = DLAPY2( DBLE( A( K-1, K ) ),
610: $ DIMAG( A( K-1, K ) ) )
1.5 ! bertrand 611: D11 = DBLE( A( K, K ) / D )
! 612: D22 = DBLE( A( K-1, K-1 ) / D )
1.1 bertrand 613: D12 = A( K-1, K ) / D
614: TT = ONE / ( D11*D22-ONE )
615: *
616: DO 30 J = K - 2, 1, -1
617: *
618: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
619: *
620: WKM1 = TT*( D11*A( J, K-1 )-DCONJG( D12 )*
621: $ A( J, K ) )
622: WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) )
623: *
624: * Perform a rank-2 update of A(1:k-2,1:k-2)
625: *
626: DO 20 I = J, 1, -1
627: A( I, J ) = A( I, J ) -
628: $ ( A( I, K ) / D )*DCONJG( WK ) -
629: $ ( A( I, K-1 ) / D )*DCONJG( WKM1 )
630: 20 CONTINUE
631: *
632: * Store U(k) and U(k-1) in cols k and k-1 for row J
633: *
634: A( J, K ) = WK / D
635: A( J, K-1 ) = WKM1 / D
636: * (*) Make sure that diagonal element of pivot is real
637: A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO )
638: *
639: 30 CONTINUE
640: *
641: END IF
642: *
643: * Copy superdiagonal elements of D(K) to E(K) and
644: * ZERO out superdiagonal entry of A
645: *
646: E( K ) = A( K-1, K )
647: E( K-1 ) = CZERO
648: A( K-1, K ) = CZERO
649: *
650: END IF
651: *
652: * End column K is nonsingular
653: *
654: END IF
655: *
656: * Store details of the interchanges in IPIV
657: *
658: IF( KSTEP.EQ.1 ) THEN
659: IPIV( K ) = KP
660: ELSE
661: IPIV( K ) = -P
662: IPIV( K-1 ) = -KP
663: END IF
664: *
665: * Decrease K and return to the start of the main loop
666: *
667: K = K - KSTEP
668: GO TO 10
669: *
670: 34 CONTINUE
671: *
672: ELSE
673: *
674: * Factorize A as L*D*L**H using the lower triangle of A
675: *
1.4 bertrand 676: * Initialize the unused last entry of the subdiagonal array E.
1.1 bertrand 677: *
678: E( N ) = CZERO
679: *
680: * K is the main loop index, increasing from 1 to N in steps of
681: * 1 or 2
682: *
683: K = 1
684: 40 CONTINUE
685: *
686: * If K > N, exit from loop
687: *
688: IF( K.GT.N )
689: $ GO TO 64
690: KSTEP = 1
691: P = K
692: *
693: * Determine rows and columns to be interchanged and whether
694: * a 1-by-1 or 2-by-2 pivot block will be used
695: *
696: ABSAKK = ABS( DBLE( A( K, K ) ) )
697: *
698: * IMAX is the row-index of the largest off-diagonal element in
699: * column K, and COLMAX is its absolute value.
700: * Determine both COLMAX and IMAX.
701: *
702: IF( K.LT.N ) THEN
703: IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 )
704: COLMAX = CABS1( A( IMAX, K ) )
705: ELSE
706: COLMAX = ZERO
707: END IF
708: *
709: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
710: *
711: * Column K is zero or underflow: set INFO and continue
712: *
713: IF( INFO.EQ.0 )
714: $ INFO = K
715: KP = K
716: A( K, K ) = DBLE( A( K, K ) )
717: *
718: * Set E( K ) to zero
719: *
720: IF( K.LT.N )
721: $ E( K ) = CZERO
722: *
723: ELSE
724: *
725: * ============================================================
726: *
727: * BEGIN pivot search
728: *
729: * Case(1)
730: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
731: * (used to handle NaN and Inf)
732: *
733: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
734: *
735: * no interchange, use 1-by-1 pivot block
736: *
737: KP = K
738: *
739: ELSE
740: *
741: DONE = .FALSE.
742: *
743: * Loop until pivot found
744: *
745: 42 CONTINUE
746: *
747: * BEGIN pivot search loop body
748: *
749: *
750: * JMAX is the column-index of the largest off-diagonal
751: * element in row IMAX, and ROWMAX is its absolute value.
752: * Determine both ROWMAX and JMAX.
753: *
754: IF( IMAX.NE.K ) THEN
755: JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA )
756: ROWMAX = CABS1( A( IMAX, JMAX ) )
757: ELSE
758: ROWMAX = ZERO
759: END IF
760: *
761: IF( IMAX.LT.N ) THEN
762: ITEMP = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ),
763: $ 1 )
764: DTEMP = CABS1( A( ITEMP, IMAX ) )
765: IF( DTEMP.GT.ROWMAX ) THEN
766: ROWMAX = DTEMP
767: JMAX = ITEMP
768: END IF
769: END IF
770: *
771: * Case(2)
772: * Equivalent to testing for
1.5 ! bertrand 773: * ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
1.1 bertrand 774: * (used to handle NaN and Inf)
775: *
776: IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) )
777: $ .LT.ALPHA*ROWMAX ) ) THEN
778: *
779: * interchange rows and columns K and IMAX,
780: * use 1-by-1 pivot block
781: *
782: KP = IMAX
783: DONE = .TRUE.
784: *
785: * Case(3)
786: * Equivalent to testing for ROWMAX.EQ.COLMAX,
787: * (used to handle NaN and Inf)
788: *
789: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
790: $ THEN
791: *
792: * interchange rows and columns K+1 and IMAX,
793: * use 2-by-2 pivot block
794: *
795: KP = IMAX
796: KSTEP = 2
797: DONE = .TRUE.
798: *
799: * Case(4)
800: ELSE
801: *
802: * Pivot not found: set params and repeat
803: *
804: P = IMAX
805: COLMAX = ROWMAX
806: IMAX = JMAX
807: END IF
808: *
809: *
810: * END pivot search loop body
811: *
812: IF( .NOT.DONE ) GOTO 42
813: *
814: END IF
815: *
816: * END pivot search
817: *
818: * ============================================================
819: *
820: * KK is the column of A where pivoting step stopped
821: *
822: KK = K + KSTEP - 1
823: *
824: * For only a 2x2 pivot, interchange rows and columns K and P
825: * in the trailing submatrix A(k:n,k:n)
826: *
827: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
828: * (1) Swap columnar parts
829: IF( P.LT.N )
830: $ CALL ZSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
831: * (2) Swap and conjugate middle parts
832: DO 44 J = K + 1, P - 1
833: T = DCONJG( A( J, K ) )
834: A( J, K ) = DCONJG( A( P, J ) )
835: A( P, J ) = T
836: 44 CONTINUE
837: * (3) Swap and conjugate corner elements at row-col interserction
838: A( P, K ) = DCONJG( A( P, K ) )
839: * (4) Swap diagonal elements at row-col intersection
840: R1 = DBLE( A( K, K ) )
841: A( K, K ) = DBLE( A( P, P ) )
842: A( P, P ) = R1
843: *
844: * Convert lower triangle of A into L form by applying
845: * the interchanges in columns 1:k-1.
846: *
847: IF ( K.GT.1 )
848: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
849: *
850: END IF
851: *
852: * For both 1x1 and 2x2 pivots, interchange rows and
853: * columns KK and KP in the trailing submatrix A(k:n,k:n)
854: *
855: IF( KP.NE.KK ) THEN
856: * (1) Swap columnar parts
857: IF( KP.LT.N )
858: $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
859: * (2) Swap and conjugate middle parts
860: DO 45 J = KK + 1, KP - 1
861: T = DCONJG( A( J, KK ) )
862: A( J, KK ) = DCONJG( A( KP, J ) )
863: A( KP, J ) = T
864: 45 CONTINUE
865: * (3) Swap and conjugate corner elements at row-col interserction
866: A( KP, KK ) = DCONJG( A( KP, KK ) )
867: * (4) Swap diagonal elements at row-col intersection
868: R1 = DBLE( A( KK, KK ) )
869: A( KK, KK ) = DBLE( A( KP, KP ) )
870: A( KP, KP ) = R1
871: *
872: IF( KSTEP.EQ.2 ) THEN
873: * (*) Make sure that diagonal element of pivot is real
874: A( K, K ) = DBLE( A( K, K ) )
875: * (5) Swap row elements
876: T = A( K+1, K )
877: A( K+1, K ) = A( KP, K )
878: A( KP, K ) = T
879: END IF
880: *
881: * Convert lower triangle of A into L form by applying
882: * the interchanges in columns 1:k-1.
883: *
884: IF ( K.GT.1 )
885: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
886: *
887: ELSE
888: * (*) Make sure that diagonal element of pivot is real
889: A( K, K ) = DBLE( A( K, K ) )
890: IF( KSTEP.EQ.2 )
891: $ A( K+1, K+1 ) = DBLE( A( K+1, K+1 ) )
892: END IF
893: *
894: * Update the trailing submatrix
895: *
896: IF( KSTEP.EQ.1 ) THEN
897: *
898: * 1-by-1 pivot block D(k): column k of A now holds
899: *
900: * W(k) = L(k)*D(k),
901: *
902: * where L(k) is the k-th column of L
903: *
904: IF( K.LT.N ) THEN
905: *
906: * Perform a rank-1 update of A(k+1:n,k+1:n) and
907: * store L(k) in column k
908: *
909: * Handle division by a small number
910: *
911: IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN
912: *
913: * Perform a rank-1 update of A(k+1:n,k+1:n) as
914: * A := A - L(k)*D(k)*L(k)**T
915: * = A - W(k)*(1/D(k))*W(k)**T
916: *
917: D11 = ONE / DBLE( A( K, K ) )
918: CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1,
919: $ A( K+1, K+1 ), LDA )
920: *
921: * Store L(k) in column k
922: *
923: CALL ZDSCAL( N-K, D11, A( K+1, K ), 1 )
924: ELSE
925: *
926: * Store L(k) in column k
927: *
928: D11 = DBLE( A( K, K ) )
929: DO 46 II = K + 1, N
930: A( II, K ) = A( II, K ) / D11
931: 46 CONTINUE
932: *
933: * Perform a rank-1 update of A(k+1:n,k+1:n) as
934: * A := A - L(k)*D(k)*L(k)**T
935: * = A - W(k)*(1/D(k))*W(k)**T
936: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
937: *
938: CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1,
939: $ A( K+1, K+1 ), LDA )
940: END IF
941: *
942: * Store the subdiagonal element of D in array E
943: *
944: E( K ) = CZERO
945: *
946: END IF
947: *
948: ELSE
949: *
950: * 2-by-2 pivot block D(k): columns k and k+1 now hold
951: *
952: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
953: *
954: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
955: * of L
956: *
957: *
958: * Perform a rank-2 update of A(k+2:n,k+2:n) as
959: *
960: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
961: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
962: *
963: * and store L(k) and L(k+1) in columns k and k+1
964: *
965: IF( K.LT.N-1 ) THEN
966: * D = |A21|
967: D = DLAPY2( DBLE( A( K+1, K ) ),
968: $ DIMAG( A( K+1, K ) ) )
969: D11 = DBLE( A( K+1, K+1 ) ) / D
970: D22 = DBLE( A( K, K ) ) / D
971: D21 = A( K+1, K ) / D
972: TT = ONE / ( D11*D22-ONE )
973: *
974: DO 60 J = K + 2, N
975: *
976: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
977: *
978: WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) )
979: WKP1 = TT*( D22*A( J, K+1 )-DCONJG( D21 )*
980: $ A( J, K ) )
981: *
982: * Perform a rank-2 update of A(k+2:n,k+2:n)
983: *
984: DO 50 I = J, N
985: A( I, J ) = A( I, J ) -
986: $ ( A( I, K ) / D )*DCONJG( WK ) -
987: $ ( A( I, K+1 ) / D )*DCONJG( WKP1 )
988: 50 CONTINUE
989: *
990: * Store L(k) and L(k+1) in cols k and k+1 for row J
991: *
992: A( J, K ) = WK / D
993: A( J, K+1 ) = WKP1 / D
994: * (*) Make sure that diagonal element of pivot is real
995: A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO )
996: *
997: 60 CONTINUE
998: *
999: END IF
1000: *
1001: * Copy subdiagonal elements of D(K) to E(K) and
1002: * ZERO out subdiagonal entry of A
1003: *
1004: E( K ) = A( K+1, K )
1005: E( K+1 ) = CZERO
1006: A( K+1, K ) = CZERO
1007: *
1008: END IF
1009: *
1010: * End column K is nonsingular
1011: *
1012: END IF
1013: *
1014: * Store details of the interchanges in IPIV
1015: *
1016: IF( KSTEP.EQ.1 ) THEN
1017: IPIV( K ) = KP
1018: ELSE
1019: IPIV( K ) = -P
1020: IPIV( K+1 ) = -KP
1021: END IF
1022: *
1023: * Increase K and return to the start of the main loop
1024: *
1025: K = K + KSTEP
1026: GO TO 40
1027: *
1028: 64 CONTINUE
1029: *
1030: END IF
1031: *
1032: RETURN
1033: *
1034: * End of ZHETF2_RK
1035: *
1036: END