1 SUBROUTINE ZSPTRF( UPLO, N, AP, IPIV, INFO )
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER INFO, N
11 * ..
12 * .. Array Arguments ..
13 INTEGER IPIV( * )
14 COMPLEX*16 AP( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZSPTRF computes the factorization of a complex symmetric matrix A
21 * stored in packed format using the Bunch-Kaufman diagonal pivoting
22 * method:
23 *
24 * A = U*D*U**T or A = L*D*L**T
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is symmetric and block diagonal with
28 * 1-by-1 and 2-by-2 diagonal blocks.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * = 'U': Upper triangle of A is stored;
35 * = 'L': Lower triangle of A is stored.
36 *
37 * N (input) INTEGER
38 * The order of the matrix A. N >= 0.
39 *
40 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
41 * On entry, the upper or lower triangle of the symmetric matrix
42 * A, packed columnwise in a linear array. The j-th column of A
43 * is stored in the array AP as follows:
44 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
45 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
46 *
47 * On exit, the block diagonal matrix D and the multipliers used
48 * to obtain the factor U or L, stored as a packed triangular
49 * matrix overwriting A (see below for further details).
50 *
51 * IPIV (output) INTEGER array, dimension (N)
52 * Details of the interchanges and the block structure of D.
53 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
54 * interchanged and D(k,k) is a 1-by-1 diagonal block.
55 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
56 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
57 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
58 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
59 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
60 *
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
65 * has been completed, but the block diagonal matrix D is
66 * exactly singular, and division by zero will occur if it
67 * is used to solve a system of equations.
68 *
69 * Further Details
70 * ===============
71 *
72 * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
73 * Company
74 *
75 * If UPLO = 'U', then A = U*D*U**T, where
76 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
77 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
78 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
79 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
80 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
81 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
82 *
83 * ( I v 0 ) k-s
84 * U(k) = ( 0 I 0 ) s
85 * ( 0 0 I ) n-k
86 * k-s s n-k
87 *
88 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
89 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
90 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
91 *
92 * If UPLO = 'L', then A = L*D*L**T, where
93 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
94 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
95 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
96 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
97 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
98 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
99 *
100 * ( I 0 0 ) k-1
101 * L(k) = ( 0 I 0 ) s
102 * ( 0 v I ) n-k-s+1
103 * k-1 s n-k-s+1
104 *
105 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
106 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
107 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
108 *
109 * =====================================================================
110 *
111 * .. Parameters ..
112 DOUBLE PRECISION ZERO, ONE
113 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
114 DOUBLE PRECISION EIGHT, SEVTEN
115 PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
116 COMPLEX*16 CONE
117 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
118 * ..
119 * .. Local Scalars ..
120 LOGICAL UPPER
121 INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
122 $ KSTEP, KX, NPP
123 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
124 COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, ZDUM
125 * ..
126 * .. External Functions ..
127 LOGICAL LSAME
128 INTEGER IZAMAX
129 EXTERNAL LSAME, IZAMAX
130 * ..
131 * .. External Subroutines ..
132 EXTERNAL XERBLA, ZSCAL, ZSPR, ZSWAP
133 * ..
134 * .. Intrinsic Functions ..
135 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
136 * ..
137 * .. Statement Functions ..
138 DOUBLE PRECISION CABS1
139 * ..
140 * .. Statement Function definitions ..
141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
142 * ..
143 * .. Executable Statements ..
144 *
145 * Test the input parameters.
146 *
147 INFO = 0
148 UPPER = LSAME( UPLO, 'U' )
149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
150 INFO = -1
151 ELSE IF( N.LT.0 ) THEN
152 INFO = -2
153 END IF
154 IF( INFO.NE.0 ) THEN
155 CALL XERBLA( 'ZSPTRF', -INFO )
156 RETURN
157 END IF
158 *
159 * Initialize ALPHA for use in choosing pivot block size.
160 *
161 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
162 *
163 IF( UPPER ) THEN
164 *
165 * Factorize A as U*D*U**T using the upper triangle of A
166 *
167 * K is the main loop index, decreasing from N to 1 in steps of
168 * 1 or 2
169 *
170 K = N
171 KC = ( N-1 )*N / 2 + 1
172 10 CONTINUE
173 KNC = KC
174 *
175 * If K < 1, exit from loop
176 *
177 IF( K.LT.1 )
178 $ GO TO 110
179 KSTEP = 1
180 *
181 * Determine rows and columns to be interchanged and whether
182 * a 1-by-1 or 2-by-2 pivot block will be used
183 *
184 ABSAKK = CABS1( AP( KC+K-1 ) )
185 *
186 * IMAX is the row-index of the largest off-diagonal element in
187 * column K, and COLMAX is its absolute value
188 *
189 IF( K.GT.1 ) THEN
190 IMAX = IZAMAX( K-1, AP( KC ), 1 )
191 COLMAX = CABS1( AP( KC+IMAX-1 ) )
192 ELSE
193 COLMAX = ZERO
194 END IF
195 *
196 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
197 *
198 * Column K is zero: set INFO and continue
199 *
200 IF( INFO.EQ.0 )
201 $ INFO = K
202 KP = K
203 ELSE
204 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
205 *
206 * no interchange, use 1-by-1 pivot block
207 *
208 KP = K
209 ELSE
210 *
211 ROWMAX = ZERO
212 JMAX = IMAX
213 KX = IMAX*( IMAX+1 ) / 2 + IMAX
214 DO 20 J = IMAX + 1, K
215 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
216 ROWMAX = CABS1( AP( KX ) )
217 JMAX = J
218 END IF
219 KX = KX + J
220 20 CONTINUE
221 KPC = ( IMAX-1 )*IMAX / 2 + 1
222 IF( IMAX.GT.1 ) THEN
223 JMAX = IZAMAX( IMAX-1, AP( KPC ), 1 )
224 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-1 ) ) )
225 END IF
226 *
227 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
228 *
229 * no interchange, use 1-by-1 pivot block
230 *
231 KP = K
232 ELSE IF( CABS1( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
233 *
234 * interchange rows and columns K and IMAX, use 1-by-1
235 * pivot block
236 *
237 KP = IMAX
238 ELSE
239 *
240 * interchange rows and columns K-1 and IMAX, use 2-by-2
241 * pivot block
242 *
243 KP = IMAX
244 KSTEP = 2
245 END IF
246 END IF
247 *
248 KK = K - KSTEP + 1
249 IF( KSTEP.EQ.2 )
250 $ KNC = KNC - K + 1
251 IF( KP.NE.KK ) THEN
252 *
253 * Interchange rows and columns KK and KP in the leading
254 * submatrix A(1:k,1:k)
255 *
256 CALL ZSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
257 KX = KPC + KP - 1
258 DO 30 J = KP + 1, KK - 1
259 KX = KX + J - 1
260 T = AP( KNC+J-1 )
261 AP( KNC+J-1 ) = AP( KX )
262 AP( KX ) = T
263 30 CONTINUE
264 T = AP( KNC+KK-1 )
265 AP( KNC+KK-1 ) = AP( KPC+KP-1 )
266 AP( KPC+KP-1 ) = T
267 IF( KSTEP.EQ.2 ) THEN
268 T = AP( KC+K-2 )
269 AP( KC+K-2 ) = AP( KC+KP-1 )
270 AP( KC+KP-1 ) = T
271 END IF
272 END IF
273 *
274 * Update the leading submatrix
275 *
276 IF( KSTEP.EQ.1 ) THEN
277 *
278 * 1-by-1 pivot block D(k): column k now holds
279 *
280 * W(k) = U(k)*D(k)
281 *
282 * where U(k) is the k-th column of U
283 *
284 * Perform a rank-1 update of A(1:k-1,1:k-1) as
285 *
286 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
287 *
288 R1 = CONE / AP( KC+K-1 )
289 CALL ZSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
290 *
291 * Store U(k) in column k
292 *
293 CALL ZSCAL( K-1, R1, AP( KC ), 1 )
294 ELSE
295 *
296 * 2-by-2 pivot block D(k): columns k and k-1 now hold
297 *
298 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
299 *
300 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
301 * of U
302 *
303 * Perform a rank-2 update of A(1:k-2,1:k-2) as
304 *
305 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
306 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
307 *
308 IF( K.GT.2 ) THEN
309 *
310 D12 = AP( K-1+( K-1 )*K / 2 )
311 D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
312 D11 = AP( K+( K-1 )*K / 2 ) / D12
313 T = CONE / ( D11*D22-CONE )
314 D12 = T / D12
315 *
316 DO 50 J = K - 2, 1, -1
317 WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
318 $ AP( J+( K-1 )*K / 2 ) )
319 WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
320 $ AP( J+( K-2 )*( K-1 ) / 2 ) )
321 DO 40 I = J, 1, -1
322 AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
323 $ AP( I+( K-1 )*K / 2 )*WK -
324 $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
325 40 CONTINUE
326 AP( J+( K-1 )*K / 2 ) = WK
327 AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
328 50 CONTINUE
329 *
330 END IF
331 END IF
332 END IF
333 *
334 * Store details of the interchanges in IPIV
335 *
336 IF( KSTEP.EQ.1 ) THEN
337 IPIV( K ) = KP
338 ELSE
339 IPIV( K ) = -KP
340 IPIV( K-1 ) = -KP
341 END IF
342 *
343 * Decrease K and return to the start of the main loop
344 *
345 K = K - KSTEP
346 KC = KNC - K
347 GO TO 10
348 *
349 ELSE
350 *
351 * Factorize A as L*D*L**T using the lower triangle of A
352 *
353 * K is the main loop index, increasing from 1 to N in steps of
354 * 1 or 2
355 *
356 K = 1
357 KC = 1
358 NPP = N*( N+1 ) / 2
359 60 CONTINUE
360 KNC = KC
361 *
362 * If K > N, exit from loop
363 *
364 IF( K.GT.N )
365 $ GO TO 110
366 KSTEP = 1
367 *
368 * Determine rows and columns to be interchanged and whether
369 * a 1-by-1 or 2-by-2 pivot block will be used
370 *
371 ABSAKK = CABS1( AP( KC ) )
372 *
373 * IMAX is the row-index of the largest off-diagonal element in
374 * column K, and COLMAX is its absolute value
375 *
376 IF( K.LT.N ) THEN
377 IMAX = K + IZAMAX( N-K, AP( KC+1 ), 1 )
378 COLMAX = CABS1( AP( KC+IMAX-K ) )
379 ELSE
380 COLMAX = ZERO
381 END IF
382 *
383 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
384 *
385 * Column K is zero: set INFO and continue
386 *
387 IF( INFO.EQ.0 )
388 $ INFO = K
389 KP = K
390 ELSE
391 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
392 *
393 * no interchange, use 1-by-1 pivot block
394 *
395 KP = K
396 ELSE
397 *
398 * JMAX is the column-index of the largest off-diagonal
399 * element in row IMAX, and ROWMAX is its absolute value
400 *
401 ROWMAX = ZERO
402 KX = KC + IMAX - K
403 DO 70 J = K, IMAX - 1
404 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
405 ROWMAX = CABS1( AP( KX ) )
406 JMAX = J
407 END IF
408 KX = KX + N - J
409 70 CONTINUE
410 KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
411 IF( IMAX.LT.N ) THEN
412 JMAX = IMAX + IZAMAX( N-IMAX, AP( KPC+1 ), 1 )
413 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-IMAX ) ) )
414 END IF
415 *
416 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
417 *
418 * no interchange, use 1-by-1 pivot block
419 *
420 KP = K
421 ELSE IF( CABS1( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
422 *
423 * interchange rows and columns K and IMAX, use 1-by-1
424 * pivot block
425 *
426 KP = IMAX
427 ELSE
428 *
429 * interchange rows and columns K+1 and IMAX, use 2-by-2
430 * pivot block
431 *
432 KP = IMAX
433 KSTEP = 2
434 END IF
435 END IF
436 *
437 KK = K + KSTEP - 1
438 IF( KSTEP.EQ.2 )
439 $ KNC = KNC + N - K + 1
440 IF( KP.NE.KK ) THEN
441 *
442 * Interchange rows and columns KK and KP in the trailing
443 * submatrix A(k:n,k:n)
444 *
445 IF( KP.LT.N )
446 $ CALL ZSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
447 $ 1 )
448 KX = KNC + KP - KK
449 DO 80 J = KK + 1, KP - 1
450 KX = KX + N - J + 1
451 T = AP( KNC+J-KK )
452 AP( KNC+J-KK ) = AP( KX )
453 AP( KX ) = T
454 80 CONTINUE
455 T = AP( KNC )
456 AP( KNC ) = AP( KPC )
457 AP( KPC ) = T
458 IF( KSTEP.EQ.2 ) THEN
459 T = AP( KC+1 )
460 AP( KC+1 ) = AP( KC+KP-K )
461 AP( KC+KP-K ) = T
462 END IF
463 END IF
464 *
465 * Update the trailing submatrix
466 *
467 IF( KSTEP.EQ.1 ) THEN
468 *
469 * 1-by-1 pivot block D(k): column k now holds
470 *
471 * W(k) = L(k)*D(k)
472 *
473 * where L(k) is the k-th column of L
474 *
475 IF( K.LT.N ) THEN
476 *
477 * Perform a rank-1 update of A(k+1:n,k+1:n) as
478 *
479 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
480 *
481 R1 = CONE / AP( KC )
482 CALL ZSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
483 $ AP( KC+N-K+1 ) )
484 *
485 * Store L(k) in column K
486 *
487 CALL ZSCAL( N-K, R1, AP( KC+1 ), 1 )
488 END IF
489 ELSE
490 *
491 * 2-by-2 pivot block D(k): columns K and K+1 now hold
492 *
493 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
494 *
495 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
496 * of L
497 *
498 IF( K.LT.N-1 ) THEN
499 *
500 * Perform a rank-2 update of A(k+2:n,k+2:n) as
501 *
502 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
503 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
504 *
505 * where L(k) and L(k+1) are the k-th and (k+1)-th
506 * columns of L
507 *
508 D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
509 D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
510 D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
511 T = CONE / ( D11*D22-CONE )
512 D21 = T / D21
513 *
514 DO 100 J = K + 2, N
515 WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
516 $ AP( J+K*( 2*N-K-1 ) / 2 ) )
517 WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
518 $ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
519 DO 90 I = J, N
520 AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
521 $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
522 $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
523 90 CONTINUE
524 AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
525 AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
526 100 CONTINUE
527 END IF
528 END IF
529 END IF
530 *
531 * Store details of the interchanges in IPIV
532 *
533 IF( KSTEP.EQ.1 ) THEN
534 IPIV( K ) = KP
535 ELSE
536 IPIV( K ) = -KP
537 IPIV( K+1 ) = -KP
538 END IF
539 *
540 * Increase K and return to the start of the main loop
541 *
542 K = K + KSTEP
543 KC = KNC + N - K + 2
544 GO TO 60
545 *
546 END IF
547 *
548 110 CONTINUE
549 RETURN
550 *
551 * End of ZSPTRF
552 *
553 END
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER INFO, N
11 * ..
12 * .. Array Arguments ..
13 INTEGER IPIV( * )
14 COMPLEX*16 AP( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZSPTRF computes the factorization of a complex symmetric matrix A
21 * stored in packed format using the Bunch-Kaufman diagonal pivoting
22 * method:
23 *
24 * A = U*D*U**T or A = L*D*L**T
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is symmetric and block diagonal with
28 * 1-by-1 and 2-by-2 diagonal blocks.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * = 'U': Upper triangle of A is stored;
35 * = 'L': Lower triangle of A is stored.
36 *
37 * N (input) INTEGER
38 * The order of the matrix A. N >= 0.
39 *
40 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
41 * On entry, the upper or lower triangle of the symmetric matrix
42 * A, packed columnwise in a linear array. The j-th column of A
43 * is stored in the array AP as follows:
44 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
45 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
46 *
47 * On exit, the block diagonal matrix D and the multipliers used
48 * to obtain the factor U or L, stored as a packed triangular
49 * matrix overwriting A (see below for further details).
50 *
51 * IPIV (output) INTEGER array, dimension (N)
52 * Details of the interchanges and the block structure of D.
53 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
54 * interchanged and D(k,k) is a 1-by-1 diagonal block.
55 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
56 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
57 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
58 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
59 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
60 *
61 * INFO (output) INTEGER
62 * = 0: successful exit
63 * < 0: if INFO = -i, the i-th argument had an illegal value
64 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
65 * has been completed, but the block diagonal matrix D is
66 * exactly singular, and division by zero will occur if it
67 * is used to solve a system of equations.
68 *
69 * Further Details
70 * ===============
71 *
72 * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
73 * Company
74 *
75 * If UPLO = 'U', then A = U*D*U**T, where
76 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
77 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
78 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
79 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
80 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
81 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
82 *
83 * ( I v 0 ) k-s
84 * U(k) = ( 0 I 0 ) s
85 * ( 0 0 I ) n-k
86 * k-s s n-k
87 *
88 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
89 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
90 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
91 *
92 * If UPLO = 'L', then A = L*D*L**T, where
93 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
94 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
95 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
96 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
97 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
98 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
99 *
100 * ( I 0 0 ) k-1
101 * L(k) = ( 0 I 0 ) s
102 * ( 0 v I ) n-k-s+1
103 * k-1 s n-k-s+1
104 *
105 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
106 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
107 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
108 *
109 * =====================================================================
110 *
111 * .. Parameters ..
112 DOUBLE PRECISION ZERO, ONE
113 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
114 DOUBLE PRECISION EIGHT, SEVTEN
115 PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
116 COMPLEX*16 CONE
117 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
118 * ..
119 * .. Local Scalars ..
120 LOGICAL UPPER
121 INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
122 $ KSTEP, KX, NPP
123 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
124 COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, ZDUM
125 * ..
126 * .. External Functions ..
127 LOGICAL LSAME
128 INTEGER IZAMAX
129 EXTERNAL LSAME, IZAMAX
130 * ..
131 * .. External Subroutines ..
132 EXTERNAL XERBLA, ZSCAL, ZSPR, ZSWAP
133 * ..
134 * .. Intrinsic Functions ..
135 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
136 * ..
137 * .. Statement Functions ..
138 DOUBLE PRECISION CABS1
139 * ..
140 * .. Statement Function definitions ..
141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
142 * ..
143 * .. Executable Statements ..
144 *
145 * Test the input parameters.
146 *
147 INFO = 0
148 UPPER = LSAME( UPLO, 'U' )
149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
150 INFO = -1
151 ELSE IF( N.LT.0 ) THEN
152 INFO = -2
153 END IF
154 IF( INFO.NE.0 ) THEN
155 CALL XERBLA( 'ZSPTRF', -INFO )
156 RETURN
157 END IF
158 *
159 * Initialize ALPHA for use in choosing pivot block size.
160 *
161 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
162 *
163 IF( UPPER ) THEN
164 *
165 * Factorize A as U*D*U**T using the upper triangle of A
166 *
167 * K is the main loop index, decreasing from N to 1 in steps of
168 * 1 or 2
169 *
170 K = N
171 KC = ( N-1 )*N / 2 + 1
172 10 CONTINUE
173 KNC = KC
174 *
175 * If K < 1, exit from loop
176 *
177 IF( K.LT.1 )
178 $ GO TO 110
179 KSTEP = 1
180 *
181 * Determine rows and columns to be interchanged and whether
182 * a 1-by-1 or 2-by-2 pivot block will be used
183 *
184 ABSAKK = CABS1( AP( KC+K-1 ) )
185 *
186 * IMAX is the row-index of the largest off-diagonal element in
187 * column K, and COLMAX is its absolute value
188 *
189 IF( K.GT.1 ) THEN
190 IMAX = IZAMAX( K-1, AP( KC ), 1 )
191 COLMAX = CABS1( AP( KC+IMAX-1 ) )
192 ELSE
193 COLMAX = ZERO
194 END IF
195 *
196 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
197 *
198 * Column K is zero: set INFO and continue
199 *
200 IF( INFO.EQ.0 )
201 $ INFO = K
202 KP = K
203 ELSE
204 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
205 *
206 * no interchange, use 1-by-1 pivot block
207 *
208 KP = K
209 ELSE
210 *
211 ROWMAX = ZERO
212 JMAX = IMAX
213 KX = IMAX*( IMAX+1 ) / 2 + IMAX
214 DO 20 J = IMAX + 1, K
215 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
216 ROWMAX = CABS1( AP( KX ) )
217 JMAX = J
218 END IF
219 KX = KX + J
220 20 CONTINUE
221 KPC = ( IMAX-1 )*IMAX / 2 + 1
222 IF( IMAX.GT.1 ) THEN
223 JMAX = IZAMAX( IMAX-1, AP( KPC ), 1 )
224 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-1 ) ) )
225 END IF
226 *
227 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
228 *
229 * no interchange, use 1-by-1 pivot block
230 *
231 KP = K
232 ELSE IF( CABS1( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
233 *
234 * interchange rows and columns K and IMAX, use 1-by-1
235 * pivot block
236 *
237 KP = IMAX
238 ELSE
239 *
240 * interchange rows and columns K-1 and IMAX, use 2-by-2
241 * pivot block
242 *
243 KP = IMAX
244 KSTEP = 2
245 END IF
246 END IF
247 *
248 KK = K - KSTEP + 1
249 IF( KSTEP.EQ.2 )
250 $ KNC = KNC - K + 1
251 IF( KP.NE.KK ) THEN
252 *
253 * Interchange rows and columns KK and KP in the leading
254 * submatrix A(1:k,1:k)
255 *
256 CALL ZSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
257 KX = KPC + KP - 1
258 DO 30 J = KP + 1, KK - 1
259 KX = KX + J - 1
260 T = AP( KNC+J-1 )
261 AP( KNC+J-1 ) = AP( KX )
262 AP( KX ) = T
263 30 CONTINUE
264 T = AP( KNC+KK-1 )
265 AP( KNC+KK-1 ) = AP( KPC+KP-1 )
266 AP( KPC+KP-1 ) = T
267 IF( KSTEP.EQ.2 ) THEN
268 T = AP( KC+K-2 )
269 AP( KC+K-2 ) = AP( KC+KP-1 )
270 AP( KC+KP-1 ) = T
271 END IF
272 END IF
273 *
274 * Update the leading submatrix
275 *
276 IF( KSTEP.EQ.1 ) THEN
277 *
278 * 1-by-1 pivot block D(k): column k now holds
279 *
280 * W(k) = U(k)*D(k)
281 *
282 * where U(k) is the k-th column of U
283 *
284 * Perform a rank-1 update of A(1:k-1,1:k-1) as
285 *
286 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
287 *
288 R1 = CONE / AP( KC+K-1 )
289 CALL ZSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
290 *
291 * Store U(k) in column k
292 *
293 CALL ZSCAL( K-1, R1, AP( KC ), 1 )
294 ELSE
295 *
296 * 2-by-2 pivot block D(k): columns k and k-1 now hold
297 *
298 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
299 *
300 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
301 * of U
302 *
303 * Perform a rank-2 update of A(1:k-2,1:k-2) as
304 *
305 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
306 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
307 *
308 IF( K.GT.2 ) THEN
309 *
310 D12 = AP( K-1+( K-1 )*K / 2 )
311 D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
312 D11 = AP( K+( K-1 )*K / 2 ) / D12
313 T = CONE / ( D11*D22-CONE )
314 D12 = T / D12
315 *
316 DO 50 J = K - 2, 1, -1
317 WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
318 $ AP( J+( K-1 )*K / 2 ) )
319 WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
320 $ AP( J+( K-2 )*( K-1 ) / 2 ) )
321 DO 40 I = J, 1, -1
322 AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
323 $ AP( I+( K-1 )*K / 2 )*WK -
324 $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
325 40 CONTINUE
326 AP( J+( K-1 )*K / 2 ) = WK
327 AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
328 50 CONTINUE
329 *
330 END IF
331 END IF
332 END IF
333 *
334 * Store details of the interchanges in IPIV
335 *
336 IF( KSTEP.EQ.1 ) THEN
337 IPIV( K ) = KP
338 ELSE
339 IPIV( K ) = -KP
340 IPIV( K-1 ) = -KP
341 END IF
342 *
343 * Decrease K and return to the start of the main loop
344 *
345 K = K - KSTEP
346 KC = KNC - K
347 GO TO 10
348 *
349 ELSE
350 *
351 * Factorize A as L*D*L**T using the lower triangle of A
352 *
353 * K is the main loop index, increasing from 1 to N in steps of
354 * 1 or 2
355 *
356 K = 1
357 KC = 1
358 NPP = N*( N+1 ) / 2
359 60 CONTINUE
360 KNC = KC
361 *
362 * If K > N, exit from loop
363 *
364 IF( K.GT.N )
365 $ GO TO 110
366 KSTEP = 1
367 *
368 * Determine rows and columns to be interchanged and whether
369 * a 1-by-1 or 2-by-2 pivot block will be used
370 *
371 ABSAKK = CABS1( AP( KC ) )
372 *
373 * IMAX is the row-index of the largest off-diagonal element in
374 * column K, and COLMAX is its absolute value
375 *
376 IF( K.LT.N ) THEN
377 IMAX = K + IZAMAX( N-K, AP( KC+1 ), 1 )
378 COLMAX = CABS1( AP( KC+IMAX-K ) )
379 ELSE
380 COLMAX = ZERO
381 END IF
382 *
383 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
384 *
385 * Column K is zero: set INFO and continue
386 *
387 IF( INFO.EQ.0 )
388 $ INFO = K
389 KP = K
390 ELSE
391 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
392 *
393 * no interchange, use 1-by-1 pivot block
394 *
395 KP = K
396 ELSE
397 *
398 * JMAX is the column-index of the largest off-diagonal
399 * element in row IMAX, and ROWMAX is its absolute value
400 *
401 ROWMAX = ZERO
402 KX = KC + IMAX - K
403 DO 70 J = K, IMAX - 1
404 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
405 ROWMAX = CABS1( AP( KX ) )
406 JMAX = J
407 END IF
408 KX = KX + N - J
409 70 CONTINUE
410 KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
411 IF( IMAX.LT.N ) THEN
412 JMAX = IMAX + IZAMAX( N-IMAX, AP( KPC+1 ), 1 )
413 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-IMAX ) ) )
414 END IF
415 *
416 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
417 *
418 * no interchange, use 1-by-1 pivot block
419 *
420 KP = K
421 ELSE IF( CABS1( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
422 *
423 * interchange rows and columns K and IMAX, use 1-by-1
424 * pivot block
425 *
426 KP = IMAX
427 ELSE
428 *
429 * interchange rows and columns K+1 and IMAX, use 2-by-2
430 * pivot block
431 *
432 KP = IMAX
433 KSTEP = 2
434 END IF
435 END IF
436 *
437 KK = K + KSTEP - 1
438 IF( KSTEP.EQ.2 )
439 $ KNC = KNC + N - K + 1
440 IF( KP.NE.KK ) THEN
441 *
442 * Interchange rows and columns KK and KP in the trailing
443 * submatrix A(k:n,k:n)
444 *
445 IF( KP.LT.N )
446 $ CALL ZSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
447 $ 1 )
448 KX = KNC + KP - KK
449 DO 80 J = KK + 1, KP - 1
450 KX = KX + N - J + 1
451 T = AP( KNC+J-KK )
452 AP( KNC+J-KK ) = AP( KX )
453 AP( KX ) = T
454 80 CONTINUE
455 T = AP( KNC )
456 AP( KNC ) = AP( KPC )
457 AP( KPC ) = T
458 IF( KSTEP.EQ.2 ) THEN
459 T = AP( KC+1 )
460 AP( KC+1 ) = AP( KC+KP-K )
461 AP( KC+KP-K ) = T
462 END IF
463 END IF
464 *
465 * Update the trailing submatrix
466 *
467 IF( KSTEP.EQ.1 ) THEN
468 *
469 * 1-by-1 pivot block D(k): column k now holds
470 *
471 * W(k) = L(k)*D(k)
472 *
473 * where L(k) is the k-th column of L
474 *
475 IF( K.LT.N ) THEN
476 *
477 * Perform a rank-1 update of A(k+1:n,k+1:n) as
478 *
479 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
480 *
481 R1 = CONE / AP( KC )
482 CALL ZSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
483 $ AP( KC+N-K+1 ) )
484 *
485 * Store L(k) in column K
486 *
487 CALL ZSCAL( N-K, R1, AP( KC+1 ), 1 )
488 END IF
489 ELSE
490 *
491 * 2-by-2 pivot block D(k): columns K and K+1 now hold
492 *
493 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
494 *
495 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
496 * of L
497 *
498 IF( K.LT.N-1 ) THEN
499 *
500 * Perform a rank-2 update of A(k+2:n,k+2:n) as
501 *
502 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
503 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
504 *
505 * where L(k) and L(k+1) are the k-th and (k+1)-th
506 * columns of L
507 *
508 D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
509 D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
510 D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
511 T = CONE / ( D11*D22-CONE )
512 D21 = T / D21
513 *
514 DO 100 J = K + 2, N
515 WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
516 $ AP( J+K*( 2*N-K-1 ) / 2 ) )
517 WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
518 $ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
519 DO 90 I = J, N
520 AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
521 $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
522 $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
523 90 CONTINUE
524 AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
525 AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
526 100 CONTINUE
527 END IF
528 END IF
529 END IF
530 *
531 * Store details of the interchanges in IPIV
532 *
533 IF( KSTEP.EQ.1 ) THEN
534 IPIV( K ) = KP
535 ELSE
536 IPIV( K ) = -KP
537 IPIV( K+1 ) = -KP
538 END IF
539 *
540 * Increase K and return to the start of the main loop
541 *
542 K = K + KSTEP
543 KC = KNC + N - K + 2
544 GO TO 60
545 *
546 END IF
547 *
548 110 CONTINUE
549 RETURN
550 *
551 * End of ZSPTRF
552 *
553 END