1: SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, N
11: * ..
12: * .. Array Arguments ..
13: INTEGER IPIV( * )
14: DOUBLE PRECISION AP( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DSPTRF computes the factorization of a real symmetric matrix A stored
21: * in packed format using the Bunch-Kaufman diagonal pivoting method:
22: *
23: * A = U*D*U**T or A = L*D*L**T
24: *
25: * where U (or L) is a product of permutation and unit upper (lower)
26: * triangular matrices, and D is symmetric and block diagonal with
27: * 1-by-1 and 2-by-2 diagonal blocks.
28: *
29: * Arguments
30: * =========
31: *
32: * UPLO (input) CHARACTER*1
33: * = 'U': Upper triangle of A is stored;
34: * = 'L': Lower triangle of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
40: * On entry, the upper or lower triangle of the symmetric matrix
41: * A, packed columnwise in a linear array. The j-th column of A
42: * is stored in the array AP as follows:
43: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
44: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
45: *
46: * On exit, the block diagonal matrix D and the multipliers used
47: * to obtain the factor U or L, stored as a packed triangular
48: * matrix overwriting A (see below for further details).
49: *
50: * IPIV (output) INTEGER array, dimension (N)
51: * Details of the interchanges and the block structure of D.
52: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
53: * interchanged and D(k,k) is a 1-by-1 diagonal block.
54: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
55: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
56: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
57: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
58: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
59: *
60: * INFO (output) INTEGER
61: * = 0: successful exit
62: * < 0: if INFO = -i, the i-th argument had an illegal value
63: * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
64: * has been completed, but the block diagonal matrix D is
65: * exactly singular, and division by zero will occur if it
66: * is used to solve a system of equations.
67: *
68: * Further Details
69: * ===============
70: *
71: * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
72: * Company
73: *
74: * If UPLO = 'U', then A = U*D*U', where
75: * U = P(n)*U(n)* ... *P(k)U(k)* ...,
76: * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
77: * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
78: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
79: * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
80: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
81: *
82: * ( I v 0 ) k-s
83: * U(k) = ( 0 I 0 ) s
84: * ( 0 0 I ) n-k
85: * k-s s n-k
86: *
87: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
88: * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
89: * and A(k,k), and v overwrites A(1:k-2,k-1:k).
90: *
91: * If UPLO = 'L', then A = L*D*L', where
92: * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
93: * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
94: * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
95: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
96: * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
97: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
98: *
99: * ( I 0 0 ) k-1
100: * L(k) = ( 0 I 0 ) s
101: * ( 0 v I ) n-k-s+1
102: * k-1 s n-k-s+1
103: *
104: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
105: * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
106: * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
107: *
108: * =====================================================================
109: *
110: * .. Parameters ..
111: DOUBLE PRECISION ZERO, ONE
112: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
113: DOUBLE PRECISION EIGHT, SEVTEN
114: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
115: * ..
116: * .. Local Scalars ..
117: LOGICAL UPPER
118: INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
119: $ KSTEP, KX, NPP
120: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
121: $ ROWMAX, T, WK, WKM1, WKP1
122: * ..
123: * .. External Functions ..
124: LOGICAL LSAME
125: INTEGER IDAMAX
126: EXTERNAL LSAME, IDAMAX
127: * ..
128: * .. External Subroutines ..
129: EXTERNAL DSCAL, DSPR, DSWAP, XERBLA
130: * ..
131: * .. Intrinsic Functions ..
132: INTRINSIC ABS, MAX, SQRT
133: * ..
134: * .. Executable Statements ..
135: *
136: * Test the input parameters.
137: *
138: INFO = 0
139: UPPER = LSAME( UPLO, 'U' )
140: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
141: INFO = -1
142: ELSE IF( N.LT.0 ) THEN
143: INFO = -2
144: END IF
145: IF( INFO.NE.0 ) THEN
146: CALL XERBLA( 'DSPTRF', -INFO )
147: RETURN
148: END IF
149: *
150: * Initialize ALPHA for use in choosing pivot block size.
151: *
152: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
153: *
154: IF( UPPER ) THEN
155: *
156: * Factorize A as U*D*U' using the upper triangle of A
157: *
158: * K is the main loop index, decreasing from N to 1 in steps of
159: * 1 or 2
160: *
161: K = N
162: KC = ( N-1 )*N / 2 + 1
163: 10 CONTINUE
164: KNC = KC
165: *
166: * If K < 1, exit from loop
167: *
168: IF( K.LT.1 )
169: $ GO TO 110
170: KSTEP = 1
171: *
172: * Determine rows and columns to be interchanged and whether
173: * a 1-by-1 or 2-by-2 pivot block will be used
174: *
175: ABSAKK = ABS( AP( KC+K-1 ) )
176: *
177: * IMAX is the row-index of the largest off-diagonal element in
178: * column K, and COLMAX is its absolute value
179: *
180: IF( K.GT.1 ) THEN
181: IMAX = IDAMAX( K-1, AP( KC ), 1 )
182: COLMAX = ABS( AP( KC+IMAX-1 ) )
183: ELSE
184: COLMAX = ZERO
185: END IF
186: *
187: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
188: *
189: * Column K is zero: set INFO and continue
190: *
191: IF( INFO.EQ.0 )
192: $ INFO = K
193: KP = K
194: ELSE
195: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
196: *
197: * no interchange, use 1-by-1 pivot block
198: *
199: KP = K
200: ELSE
201: *
202: * JMAX is the column-index of the largest off-diagonal
203: * element in row IMAX, and ROWMAX is its absolute value
204: *
205: ROWMAX = ZERO
206: JMAX = IMAX
207: KX = IMAX*( IMAX+1 ) / 2 + IMAX
208: DO 20 J = IMAX + 1, K
209: IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
210: ROWMAX = ABS( AP( KX ) )
211: JMAX = J
212: END IF
213: KX = KX + J
214: 20 CONTINUE
215: KPC = ( IMAX-1 )*IMAX / 2 + 1
216: IF( IMAX.GT.1 ) THEN
217: JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 )
218: ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) )
219: END IF
220: *
221: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
222: *
223: * no interchange, use 1-by-1 pivot block
224: *
225: KP = K
226: ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
227: *
228: * interchange rows and columns K and IMAX, use 1-by-1
229: * pivot block
230: *
231: KP = IMAX
232: ELSE
233: *
234: * interchange rows and columns K-1 and IMAX, use 2-by-2
235: * pivot block
236: *
237: KP = IMAX
238: KSTEP = 2
239: END IF
240: END IF
241: *
242: KK = K - KSTEP + 1
243: IF( KSTEP.EQ.2 )
244: $ KNC = KNC - K + 1
245: IF( KP.NE.KK ) THEN
246: *
247: * Interchange rows and columns KK and KP in the leading
248: * submatrix A(1:k,1:k)
249: *
250: CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
251: KX = KPC + KP - 1
252: DO 30 J = KP + 1, KK - 1
253: KX = KX + J - 1
254: T = AP( KNC+J-1 )
255: AP( KNC+J-1 ) = AP( KX )
256: AP( KX ) = T
257: 30 CONTINUE
258: T = AP( KNC+KK-1 )
259: AP( KNC+KK-1 ) = AP( KPC+KP-1 )
260: AP( KPC+KP-1 ) = T
261: IF( KSTEP.EQ.2 ) THEN
262: T = AP( KC+K-2 )
263: AP( KC+K-2 ) = AP( KC+KP-1 )
264: AP( KC+KP-1 ) = T
265: END IF
266: END IF
267: *
268: * Update the leading submatrix
269: *
270: IF( KSTEP.EQ.1 ) THEN
271: *
272: * 1-by-1 pivot block D(k): column k now holds
273: *
274: * W(k) = U(k)*D(k)
275: *
276: * where U(k) is the k-th column of U
277: *
278: * Perform a rank-1 update of A(1:k-1,1:k-1) as
279: *
280: * A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
281: *
282: R1 = ONE / AP( KC+K-1 )
283: CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
284: *
285: * Store U(k) in column k
286: *
287: CALL DSCAL( K-1, R1, AP( KC ), 1 )
288: ELSE
289: *
290: * 2-by-2 pivot block D(k): columns k and k-1 now hold
291: *
292: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
293: *
294: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
295: * of U
296: *
297: * Perform a rank-2 update of A(1:k-2,1:k-2) as
298: *
299: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
300: * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
301: *
302: IF( K.GT.2 ) THEN
303: *
304: D12 = AP( K-1+( K-1 )*K / 2 )
305: D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
306: D11 = AP( K+( K-1 )*K / 2 ) / D12
307: T = ONE / ( D11*D22-ONE )
308: D12 = T / D12
309: *
310: DO 50 J = K - 2, 1, -1
311: WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
312: $ AP( J+( K-1 )*K / 2 ) )
313: WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
314: $ AP( J+( K-2 )*( K-1 ) / 2 ) )
315: DO 40 I = J, 1, -1
316: AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
317: $ AP( I+( K-1 )*K / 2 )*WK -
318: $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
319: 40 CONTINUE
320: AP( J+( K-1 )*K / 2 ) = WK
321: AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
322: 50 CONTINUE
323: *
324: END IF
325: *
326: END IF
327: END IF
328: *
329: * Store details of the interchanges in IPIV
330: *
331: IF( KSTEP.EQ.1 ) THEN
332: IPIV( K ) = KP
333: ELSE
334: IPIV( K ) = -KP
335: IPIV( K-1 ) = -KP
336: END IF
337: *
338: * Decrease K and return to the start of the main loop
339: *
340: K = K - KSTEP
341: KC = KNC - K
342: GO TO 10
343: *
344: ELSE
345: *
346: * Factorize A as L*D*L' using the lower triangle of A
347: *
348: * K is the main loop index, increasing from 1 to N in steps of
349: * 1 or 2
350: *
351: K = 1
352: KC = 1
353: NPP = N*( N+1 ) / 2
354: 60 CONTINUE
355: KNC = KC
356: *
357: * If K > N, exit from loop
358: *
359: IF( K.GT.N )
360: $ GO TO 110
361: KSTEP = 1
362: *
363: * Determine rows and columns to be interchanged and whether
364: * a 1-by-1 or 2-by-2 pivot block will be used
365: *
366: ABSAKK = ABS( AP( KC ) )
367: *
368: * IMAX is the row-index of the largest off-diagonal element in
369: * column K, and COLMAX is its absolute value
370: *
371: IF( K.LT.N ) THEN
372: IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 )
373: COLMAX = ABS( AP( KC+IMAX-K ) )
374: ELSE
375: COLMAX = ZERO
376: END IF
377: *
378: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
379: *
380: * Column K is zero: set INFO and continue
381: *
382: IF( INFO.EQ.0 )
383: $ INFO = K
384: KP = K
385: ELSE
386: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
387: *
388: * no interchange, use 1-by-1 pivot block
389: *
390: KP = K
391: ELSE
392: *
393: * JMAX is the column-index of the largest off-diagonal
394: * element in row IMAX, and ROWMAX is its absolute value
395: *
396: ROWMAX = ZERO
397: KX = KC + IMAX - K
398: DO 70 J = K, IMAX - 1
399: IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
400: ROWMAX = ABS( AP( KX ) )
401: JMAX = J
402: END IF
403: KX = KX + N - J
404: 70 CONTINUE
405: KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
406: IF( IMAX.LT.N ) THEN
407: JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 )
408: ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) )
409: END IF
410: *
411: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
412: *
413: * no interchange, use 1-by-1 pivot block
414: *
415: KP = K
416: ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
417: *
418: * interchange rows and columns K and IMAX, use 1-by-1
419: * pivot block
420: *
421: KP = IMAX
422: ELSE
423: *
424: * interchange rows and columns K+1 and IMAX, use 2-by-2
425: * pivot block
426: *
427: KP = IMAX
428: KSTEP = 2
429: END IF
430: END IF
431: *
432: KK = K + KSTEP - 1
433: IF( KSTEP.EQ.2 )
434: $ KNC = KNC + N - K + 1
435: IF( KP.NE.KK ) THEN
436: *
437: * Interchange rows and columns KK and KP in the trailing
438: * submatrix A(k:n,k:n)
439: *
440: IF( KP.LT.N )
441: $ CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
442: $ 1 )
443: KX = KNC + KP - KK
444: DO 80 J = KK + 1, KP - 1
445: KX = KX + N - J + 1
446: T = AP( KNC+J-KK )
447: AP( KNC+J-KK ) = AP( KX )
448: AP( KX ) = T
449: 80 CONTINUE
450: T = AP( KNC )
451: AP( KNC ) = AP( KPC )
452: AP( KPC ) = T
453: IF( KSTEP.EQ.2 ) THEN
454: T = AP( KC+1 )
455: AP( KC+1 ) = AP( KC+KP-K )
456: AP( KC+KP-K ) = T
457: END IF
458: END IF
459: *
460: * Update the trailing submatrix
461: *
462: IF( KSTEP.EQ.1 ) THEN
463: *
464: * 1-by-1 pivot block D(k): column k now holds
465: *
466: * W(k) = L(k)*D(k)
467: *
468: * where L(k) is the k-th column of L
469: *
470: IF( K.LT.N ) THEN
471: *
472: * Perform a rank-1 update of A(k+1:n,k+1:n) as
473: *
474: * A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
475: *
476: R1 = ONE / AP( KC )
477: CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
478: $ AP( KC+N-K+1 ) )
479: *
480: * Store L(k) in column K
481: *
482: CALL DSCAL( N-K, R1, AP( KC+1 ), 1 )
483: END IF
484: ELSE
485: *
486: * 2-by-2 pivot block D(k): columns K and K+1 now hold
487: *
488: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
489: *
490: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
491: * of L
492: *
493: IF( K.LT.N-1 ) THEN
494: *
495: * Perform a rank-2 update of A(k+2:n,k+2:n) as
496: *
497: * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
498: * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
499: *
500: D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
501: D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
502: D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
503: T = ONE / ( D11*D22-ONE )
504: D21 = T / D21
505: *
506: DO 100 J = K + 2, N
507: WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
508: $ AP( J+K*( 2*N-K-1 ) / 2 ) )
509: WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
510: $ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
511: *
512: DO 90 I = J, N
513: AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
514: $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
515: $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
516: 90 CONTINUE
517: *
518: AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
519: AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
520: *
521: 100 CONTINUE
522: END IF
523: END IF
524: END IF
525: *
526: * Store details of the interchanges in IPIV
527: *
528: IF( KSTEP.EQ.1 ) THEN
529: IPIV( K ) = KP
530: ELSE
531: IPIV( K ) = -KP
532: IPIV( K+1 ) = -KP
533: END IF
534: *
535: * Increase K and return to the start of the main loop
536: *
537: K = K + KSTEP
538: KC = KNC + N - K + 2
539: GO TO 60
540: *
541: END IF
542: *
543: 110 CONTINUE
544: RETURN
545: *
546: * End of DSPTRF
547: *
548: END
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