LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

◆ dsptrf()

 subroutine dsptrf ( character UPLO, integer N, double precision, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO )

DSPTRF

Purpose:
``` DSPTRF computes the factorization of a real symmetric matrix A stored
in packed format using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] AP ``` AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.```
Further Details:
```  If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    v    0   )   k-s
U(k) =  (   0    I    0   )   s
(   0    0    I   )   n-k
k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    0     0   )  k-1
L(k) =  (   0    I     0   )  s
(   0    v     I   )  n-k-s+1
k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).```
Contributors:
J. Lewis, Boeing Computer Services Company

Definition at line 158 of file dsptrf.f.

159 *
160 * -- LAPACK computational routine --
161 * -- LAPACK is a software package provided by Univ. of Tennessee, --
162 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
163 *
164 * .. Scalar Arguments ..
165  CHARACTER UPLO
166  INTEGER INFO, N
167 * ..
168 * .. Array Arguments ..
169  INTEGER IPIV( * )
170  DOUBLE PRECISION AP( * )
171 * ..
172 *
173 * =====================================================================
174 *
175 * .. Parameters ..
176  DOUBLE PRECISION ZERO, ONE
177  parameter( zero = 0.0d+0, one = 1.0d+0 )
178  DOUBLE PRECISION EIGHT, SEVTEN
179  parameter( eight = 8.0d+0, sevten = 17.0d+0 )
180 * ..
181 * .. Local Scalars ..
182  LOGICAL UPPER
183  INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
184  \$ KSTEP, KX, NPP
185  DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
186  \$ ROWMAX, T, WK, WKM1, WKP1
187 * ..
188 * .. External Functions ..
189  LOGICAL LSAME
190  INTEGER IDAMAX
191  EXTERNAL lsame, idamax
192 * ..
193 * .. External Subroutines ..
194  EXTERNAL dscal, dspr, dswap, xerbla
195 * ..
196 * .. Intrinsic Functions ..
197  INTRINSIC abs, max, sqrt
198 * ..
199 * .. Executable Statements ..
200 *
201 * Test the input parameters.
202 *
203  info = 0
204  upper = lsame( uplo, 'U' )
205  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
206  info = -1
207  ELSE IF( n.LT.0 ) THEN
208  info = -2
209  END IF
210  IF( info.NE.0 ) THEN
211  CALL xerbla( 'DSPTRF', -info )
212  RETURN
213  END IF
214 *
215 * Initialize ALPHA for use in choosing pivot block size.
216 *
217  alpha = ( one+sqrt( sevten ) ) / eight
218 *
219  IF( upper ) THEN
220 *
221 * Factorize A as U*D*U**T using the upper triangle of A
222 *
223 * K is the main loop index, decreasing from N to 1 in steps of
224 * 1 or 2
225 *
226  k = n
227  kc = ( n-1 )*n / 2 + 1
228  10 CONTINUE
229  knc = kc
230 *
231 * If K < 1, exit from loop
232 *
233  IF( k.LT.1 )
234  \$ GO TO 110
235  kstep = 1
236 *
237 * Determine rows and columns to be interchanged and whether
238 * a 1-by-1 or 2-by-2 pivot block will be used
239 *
240  absakk = abs( ap( kc+k-1 ) )
241 *
242 * IMAX is the row-index of the largest off-diagonal element in
243 * column K, and COLMAX is its absolute value
244 *
245  IF( k.GT.1 ) THEN
246  imax = idamax( k-1, ap( kc ), 1 )
247  colmax = abs( ap( kc+imax-1 ) )
248  ELSE
249  colmax = zero
250  END IF
251 *
252  IF( max( absakk, colmax ).EQ.zero ) THEN
253 *
254 * Column K is zero: set INFO and continue
255 *
256  IF( info.EQ.0 )
257  \$ info = k
258  kp = k
259  ELSE
260  IF( absakk.GE.alpha*colmax ) THEN
261 *
262 * no interchange, use 1-by-1 pivot block
263 *
264  kp = k
265  ELSE
266 *
267  rowmax = zero
268  jmax = imax
269  kx = imax*( imax+1 ) / 2 + imax
270  DO 20 j = imax + 1, k
271  IF( abs( ap( kx ) ).GT.rowmax ) THEN
272  rowmax = abs( ap( kx ) )
273  jmax = j
274  END IF
275  kx = kx + j
276  20 CONTINUE
277  kpc = ( imax-1 )*imax / 2 + 1
278  IF( imax.GT.1 ) THEN
279  jmax = idamax( imax-1, ap( kpc ), 1 )
280  rowmax = max( rowmax, abs( ap( kpc+jmax-1 ) ) )
281  END IF
282 *
283  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
284 *
285 * no interchange, use 1-by-1 pivot block
286 *
287  kp = k
288  ELSE IF( abs( ap( kpc+imax-1 ) ).GE.alpha*rowmax ) THEN
289 *
290 * interchange rows and columns K and IMAX, use 1-by-1
291 * pivot block
292 *
293  kp = imax
294  ELSE
295 *
296 * interchange rows and columns K-1 and IMAX, use 2-by-2
297 * pivot block
298 *
299  kp = imax
300  kstep = 2
301  END IF
302  END IF
303 *
304  kk = k - kstep + 1
305  IF( kstep.EQ.2 )
306  \$ knc = knc - k + 1
307  IF( kp.NE.kk ) THEN
308 *
309 * Interchange rows and columns KK and KP in the leading
310 * submatrix A(1:k,1:k)
311 *
312  CALL dswap( kp-1, ap( knc ), 1, ap( kpc ), 1 )
313  kx = kpc + kp - 1
314  DO 30 j = kp + 1, kk - 1
315  kx = kx + j - 1
316  t = ap( knc+j-1 )
317  ap( knc+j-1 ) = ap( kx )
318  ap( kx ) = t
319  30 CONTINUE
320  t = ap( knc+kk-1 )
321  ap( knc+kk-1 ) = ap( kpc+kp-1 )
322  ap( kpc+kp-1 ) = t
323  IF( kstep.EQ.2 ) THEN
324  t = ap( kc+k-2 )
325  ap( kc+k-2 ) = ap( kc+kp-1 )
326  ap( kc+kp-1 ) = t
327  END IF
328  END IF
329 *
330 * Update the leading submatrix
331 *
332  IF( kstep.EQ.1 ) THEN
333 *
334 * 1-by-1 pivot block D(k): column k now holds
335 *
336 * W(k) = U(k)*D(k)
337 *
338 * where U(k) is the k-th column of U
339 *
340 * Perform a rank-1 update of A(1:k-1,1:k-1) as
341 *
342 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
343 *
344  r1 = one / ap( kc+k-1 )
345  CALL dspr( uplo, k-1, -r1, ap( kc ), 1, ap )
346 *
347 * Store U(k) in column k
348 *
349  CALL dscal( k-1, r1, ap( kc ), 1 )
350  ELSE
351 *
352 * 2-by-2 pivot block D(k): columns k and k-1 now hold
353 *
354 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
355 *
356 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
357 * of U
358 *
359 * Perform a rank-2 update of A(1:k-2,1:k-2) as
360 *
361 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
362 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
363 *
364  IF( k.GT.2 ) THEN
365 *
366  d12 = ap( k-1+( k-1 )*k / 2 )
367  d22 = ap( k-1+( k-2 )*( k-1 ) / 2 ) / d12
368  d11 = ap( k+( k-1 )*k / 2 ) / d12
369  t = one / ( d11*d22-one )
370  d12 = t / d12
371 *
372  DO 50 j = k - 2, 1, -1
373  wkm1 = d12*( d11*ap( j+( k-2 )*( k-1 ) / 2 )-
374  \$ ap( j+( k-1 )*k / 2 ) )
375  wk = d12*( d22*ap( j+( k-1 )*k / 2 )-
376  \$ ap( j+( k-2 )*( k-1 ) / 2 ) )
377  DO 40 i = j, 1, -1
378  ap( i+( j-1 )*j / 2 ) = ap( i+( j-1 )*j / 2 ) -
379  \$ ap( i+( k-1 )*k / 2 )*wk -
380  \$ ap( i+( k-2 )*( k-1 ) / 2 )*wkm1
381  40 CONTINUE
382  ap( j+( k-1 )*k / 2 ) = wk
383  ap( j+( k-2 )*( k-1 ) / 2 ) = wkm1
384  50 CONTINUE
385 *
386  END IF
387 *
388  END IF
389  END IF
390 *
391 * Store details of the interchanges in IPIV
392 *
393  IF( kstep.EQ.1 ) THEN
394  ipiv( k ) = kp
395  ELSE
396  ipiv( k ) = -kp
397  ipiv( k-1 ) = -kp
398  END IF
399 *
400 * Decrease K and return to the start of the main loop
401 *
402  k = k - kstep
403  kc = knc - k
404  GO TO 10
405 *
406  ELSE
407 *
408 * Factorize A as L*D*L**T using the lower triangle of A
409 *
410 * K is the main loop index, increasing from 1 to N in steps of
411 * 1 or 2
412 *
413  k = 1
414  kc = 1
415  npp = n*( n+1 ) / 2
416  60 CONTINUE
417  knc = kc
418 *
419 * If K > N, exit from loop
420 *
421  IF( k.GT.n )
422  \$ GO TO 110
423  kstep = 1
424 *
425 * Determine rows and columns to be interchanged and whether
426 * a 1-by-1 or 2-by-2 pivot block will be used
427 *
428  absakk = abs( ap( kc ) )
429 *
430 * IMAX is the row-index of the largest off-diagonal element in
431 * column K, and COLMAX is its absolute value
432 *
433  IF( k.LT.n ) THEN
434  imax = k + idamax( n-k, ap( kc+1 ), 1 )
435  colmax = abs( ap( kc+imax-k ) )
436  ELSE
437  colmax = zero
438  END IF
439 *
440  IF( max( absakk, colmax ).EQ.zero ) THEN
441 *
442 * Column K is zero: set INFO and continue
443 *
444  IF( info.EQ.0 )
445  \$ info = k
446  kp = k
447  ELSE
448  IF( absakk.GE.alpha*colmax ) THEN
449 *
450 * no interchange, use 1-by-1 pivot block
451 *
452  kp = k
453  ELSE
454 *
455 * JMAX is the column-index of the largest off-diagonal
456 * element in row IMAX, and ROWMAX is its absolute value
457 *
458  rowmax = zero
459  kx = kc + imax - k
460  DO 70 j = k, imax - 1
461  IF( abs( ap( kx ) ).GT.rowmax ) THEN
462  rowmax = abs( ap( kx ) )
463  jmax = j
464  END IF
465  kx = kx + n - j
466  70 CONTINUE
467  kpc = npp - ( n-imax+1 )*( n-imax+2 ) / 2 + 1
468  IF( imax.LT.n ) THEN
469  jmax = imax + idamax( n-imax, ap( kpc+1 ), 1 )
470  rowmax = max( rowmax, abs( ap( kpc+jmax-imax ) ) )
471  END IF
472 *
473  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
474 *
475 * no interchange, use 1-by-1 pivot block
476 *
477  kp = k
478  ELSE IF( abs( ap( kpc ) ).GE.alpha*rowmax ) THEN
479 *
480 * interchange rows and columns K and IMAX, use 1-by-1
481 * pivot block
482 *
483  kp = imax
484  ELSE
485 *
486 * interchange rows and columns K+1 and IMAX, use 2-by-2
487 * pivot block
488 *
489  kp = imax
490  kstep = 2
491  END IF
492  END IF
493 *
494  kk = k + kstep - 1
495  IF( kstep.EQ.2 )
496  \$ knc = knc + n - k + 1
497  IF( kp.NE.kk ) THEN
498 *
499 * Interchange rows and columns KK and KP in the trailing
500 * submatrix A(k:n,k:n)
501 *
502  IF( kp.LT.n )
503  \$ CALL dswap( n-kp, ap( knc+kp-kk+1 ), 1, ap( kpc+1 ),
504  \$ 1 )
505  kx = knc + kp - kk
506  DO 80 j = kk + 1, kp - 1
507  kx = kx + n - j + 1
508  t = ap( knc+j-kk )
509  ap( knc+j-kk ) = ap( kx )
510  ap( kx ) = t
511  80 CONTINUE
512  t = ap( knc )
513  ap( knc ) = ap( kpc )
514  ap( kpc ) = t
515  IF( kstep.EQ.2 ) THEN
516  t = ap( kc+1 )
517  ap( kc+1 ) = ap( kc+kp-k )
518  ap( kc+kp-k ) = t
519  END IF
520  END IF
521 *
522 * Update the trailing submatrix
523 *
524  IF( kstep.EQ.1 ) THEN
525 *
526 * 1-by-1 pivot block D(k): column k now holds
527 *
528 * W(k) = L(k)*D(k)
529 *
530 * where L(k) is the k-th column of L
531 *
532  IF( k.LT.n ) THEN
533 *
534 * Perform a rank-1 update of A(k+1:n,k+1:n) as
535 *
536 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
537 *
538  r1 = one / ap( kc )
539  CALL dspr( uplo, n-k, -r1, ap( kc+1 ), 1,
540  \$ ap( kc+n-k+1 ) )
541 *
542 * Store L(k) in column K
543 *
544  CALL dscal( n-k, r1, ap( kc+1 ), 1 )
545  END IF
546  ELSE
547 *
548 * 2-by-2 pivot block D(k): columns K and K+1 now hold
549 *
550 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
551 *
552 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
553 * of L
554 *
555  IF( k.LT.n-1 ) THEN
556 *
557 * Perform a rank-2 update of A(k+2:n,k+2:n) as
558 *
559 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
560 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
561 *
562 * where L(k) and L(k+1) are the k-th and (k+1)-th
563 * columns of L
564 *
565  d21 = ap( k+1+( k-1 )*( 2*n-k ) / 2 )
566  d11 = ap( k+1+k*( 2*n-k-1 ) / 2 ) / d21
567  d22 = ap( k+( k-1 )*( 2*n-k ) / 2 ) / d21
568  t = one / ( d11*d22-one )
569  d21 = t / d21
570 *
571  DO 100 j = k + 2, n
572  wk = d21*( d11*ap( j+( k-1 )*( 2*n-k ) / 2 )-
573  \$ ap( j+k*( 2*n-k-1 ) / 2 ) )
574  wkp1 = d21*( d22*ap( j+k*( 2*n-k-1 ) / 2 )-
575  \$ ap( j+( k-1 )*( 2*n-k ) / 2 ) )
576 *
577  DO 90 i = j, n
578  ap( i+( j-1 )*( 2*n-j ) / 2 ) = ap( i+( j-1 )*
579  \$ ( 2*n-j ) / 2 ) - ap( i+( k-1 )*( 2*n-k ) /
580  \$ 2 )*wk - ap( i+k*( 2*n-k-1 ) / 2 )*wkp1
581  90 CONTINUE
582 *
583  ap( j+( k-1 )*( 2*n-k ) / 2 ) = wk
584  ap( j+k*( 2*n-k-1 ) / 2 ) = wkp1
585 *
586  100 CONTINUE
587  END IF
588  END IF
589  END IF
590 *
591 * Store details of the interchanges in IPIV
592 *
593  IF( kstep.EQ.1 ) THEN
594  ipiv( k ) = kp
595  ELSE
596  ipiv( k ) = -kp
597  ipiv( k+1 ) = -kp
598  END IF
599 *
600 * Increase K and return to the start of the main loop
601 *
602  k = k + kstep
603  kc = knc + n - k + 2
604  GO TO 60
605 *
606  END IF
607 *
608  110 CONTINUE
609  RETURN
610 *
611 * End of DSPTRF
612 *
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dspr(UPLO, N, ALPHA, X, INCX, AP)
DSPR
Definition: dspr.f:127
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