 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssptrf()

 subroutine ssptrf ( character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO )

SSPTRF

Purpose:
``` SSPTRF 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 REAL 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:
```  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company

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).```

Definition at line 156 of file ssptrf.f.

157 *
158 * -- LAPACK computational routine --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 *
162 * .. Scalar Arguments ..
163  CHARACTER UPLO
164  INTEGER INFO, N
165 * ..
166 * .. Array Arguments ..
167  INTEGER IPIV( * )
168  REAL AP( * )
169 * ..
170 *
171 * =====================================================================
172 *
173 * .. Parameters ..
174  REAL ZERO, ONE
175  parameter( zero = 0.0e+0, one = 1.0e+0 )
176  REAL EIGHT, SEVTEN
177  parameter( eight = 8.0e+0, sevten = 17.0e+0 )
178 * ..
179 * .. Local Scalars ..
180  LOGICAL UPPER
181  INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
182  \$ KSTEP, KX, NPP
183  REAL ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
184  \$ ROWMAX, T, WK, WKM1, WKP1
185 * ..
186 * .. External Functions ..
187  LOGICAL LSAME
188  INTEGER ISAMAX
189  EXTERNAL lsame, isamax
190 * ..
191 * .. External Subroutines ..
192  EXTERNAL sscal, sspr, sswap, xerbla
193 * ..
194 * .. Intrinsic Functions ..
195  INTRINSIC abs, max, sqrt
196 * ..
197 * .. Executable Statements ..
198 *
199 * Test the input parameters.
200 *
201  info = 0
202  upper = lsame( uplo, 'U' )
203  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
204  info = -1
205  ELSE IF( n.LT.0 ) THEN
206  info = -2
207  END IF
208  IF( info.NE.0 ) THEN
209  CALL xerbla( 'SSPTRF', -info )
210  RETURN
211  END IF
212 *
213 * Initialize ALPHA for use in choosing pivot block size.
214 *
215  alpha = ( one+sqrt( sevten ) ) / eight
216 *
217  IF( upper ) THEN
218 *
219 * Factorize A as U*D*U**T using the upper triangle of A
220 *
221 * K is the main loop index, decreasing from N to 1 in steps of
222 * 1 or 2
223 *
224  k = n
225  kc = ( n-1 )*n / 2 + 1
226  10 CONTINUE
227  knc = kc
228 *
229 * If K < 1, exit from loop
230 *
231  IF( k.LT.1 )
232  \$ GO TO 110
233  kstep = 1
234 *
235 * Determine rows and columns to be interchanged and whether
236 * a 1-by-1 or 2-by-2 pivot block will be used
237 *
238  absakk = abs( ap( kc+k-1 ) )
239 *
240 * IMAX is the row-index of the largest off-diagonal element in
241 * column K, and COLMAX is its absolute value
242 *
243  IF( k.GT.1 ) THEN
244  imax = isamax( k-1, ap( kc ), 1 )
245  colmax = abs( ap( kc+imax-1 ) )
246  ELSE
247  colmax = zero
248  END IF
249 *
250  IF( max( absakk, colmax ).EQ.zero ) THEN
251 *
252 * Column K is zero: set INFO and continue
253 *
254  IF( info.EQ.0 )
255  \$ info = k
256  kp = k
257  ELSE
258  IF( absakk.GE.alpha*colmax ) THEN
259 *
260 * no interchange, use 1-by-1 pivot block
261 *
262  kp = k
263  ELSE
264 *
265  rowmax = zero
266  jmax = imax
267  kx = imax*( imax+1 ) / 2 + imax
268  DO 20 j = imax + 1, k
269  IF( abs( ap( kx ) ).GT.rowmax ) THEN
270  rowmax = abs( ap( kx ) )
271  jmax = j
272  END IF
273  kx = kx + j
274  20 CONTINUE
275  kpc = ( imax-1 )*imax / 2 + 1
276  IF( imax.GT.1 ) THEN
277  jmax = isamax( imax-1, ap( kpc ), 1 )
278  rowmax = max( rowmax, abs( ap( kpc+jmax-1 ) ) )
279  END IF
280 *
281  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
282 *
283 * no interchange, use 1-by-1 pivot block
284 *
285  kp = k
286  ELSE IF( abs( ap( kpc+imax-1 ) ).GE.alpha*rowmax ) THEN
287 *
288 * interchange rows and columns K and IMAX, use 1-by-1
289 * pivot block
290 *
291  kp = imax
292  ELSE
293 *
294 * interchange rows and columns K-1 and IMAX, use 2-by-2
295 * pivot block
296 *
297  kp = imax
298  kstep = 2
299  END IF
300  END IF
301 *
302  kk = k - kstep + 1
303  IF( kstep.EQ.2 )
304  \$ knc = knc - k + 1
305  IF( kp.NE.kk ) THEN
306 *
307 * Interchange rows and columns KK and KP in the leading
308 * submatrix A(1:k,1:k)
309 *
310  CALL sswap( kp-1, ap( knc ), 1, ap( kpc ), 1 )
311  kx = kpc + kp - 1
312  DO 30 j = kp + 1, kk - 1
313  kx = kx + j - 1
314  t = ap( knc+j-1 )
315  ap( knc+j-1 ) = ap( kx )
316  ap( kx ) = t
317  30 CONTINUE
318  t = ap( knc+kk-1 )
319  ap( knc+kk-1 ) = ap( kpc+kp-1 )
320  ap( kpc+kp-1 ) = t
321  IF( kstep.EQ.2 ) THEN
322  t = ap( kc+k-2 )
323  ap( kc+k-2 ) = ap( kc+kp-1 )
324  ap( kc+kp-1 ) = t
325  END IF
326  END IF
327 *
328 * Update the leading submatrix
329 *
330  IF( kstep.EQ.1 ) THEN
331 *
332 * 1-by-1 pivot block D(k): column k now holds
333 *
334 * W(k) = U(k)*D(k)
335 *
336 * where U(k) is the k-th column of U
337 *
338 * Perform a rank-1 update of A(1:k-1,1:k-1) as
339 *
340 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
341 *
342  r1 = one / ap( kc+k-1 )
343  CALL sspr( uplo, k-1, -r1, ap( kc ), 1, ap )
344 *
345 * Store U(k) in column k
346 *
347  CALL sscal( k-1, r1, ap( kc ), 1 )
348  ELSE
349 *
350 * 2-by-2 pivot block D(k): columns k and k-1 now hold
351 *
352 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
353 *
354 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
355 * of U
356 *
357 * Perform a rank-2 update of A(1:k-2,1:k-2) as
358 *
359 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
360 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
361 *
362  IF( k.GT.2 ) THEN
363 *
364  d12 = ap( k-1+( k-1 )*k / 2 )
365  d22 = ap( k-1+( k-2 )*( k-1 ) / 2 ) / d12
366  d11 = ap( k+( k-1 )*k / 2 ) / d12
367  t = one / ( d11*d22-one )
368  d12 = t / d12
369 *
370  DO 50 j = k - 2, 1, -1
371  wkm1 = d12*( d11*ap( j+( k-2 )*( k-1 ) / 2 )-
372  \$ ap( j+( k-1 )*k / 2 ) )
373  wk = d12*( d22*ap( j+( k-1 )*k / 2 )-
374  \$ ap( j+( k-2 )*( k-1 ) / 2 ) )
375  DO 40 i = j, 1, -1
376  ap( i+( j-1 )*j / 2 ) = ap( i+( j-1 )*j / 2 ) -
377  \$ ap( i+( k-1 )*k / 2 )*wk -
378  \$ ap( i+( k-2 )*( k-1 ) / 2 )*wkm1
379  40 CONTINUE
380  ap( j+( k-1 )*k / 2 ) = wk
381  ap( j+( k-2 )*( k-1 ) / 2 ) = wkm1
382  50 CONTINUE
383 *
384  END IF
385 *
386  END IF
387  END IF
388 *
389 * Store details of the interchanges in IPIV
390 *
391  IF( kstep.EQ.1 ) THEN
392  ipiv( k ) = kp
393  ELSE
394  ipiv( k ) = -kp
395  ipiv( k-1 ) = -kp
396  END IF
397 *
398 * Decrease K and return to the start of the main loop
399 *
400  k = k - kstep
401  kc = knc - k
402  GO TO 10
403 *
404  ELSE
405 *
406 * Factorize A as L*D*L**T using the lower triangle of A
407 *
408 * K is the main loop index, increasing from 1 to N in steps of
409 * 1 or 2
410 *
411  k = 1
412  kc = 1
413  npp = n*( n+1 ) / 2
414  60 CONTINUE
415  knc = kc
416 *
417 * If K > N, exit from loop
418 *
419  IF( k.GT.n )
420  \$ GO TO 110
421  kstep = 1
422 *
423 * Determine rows and columns to be interchanged and whether
424 * a 1-by-1 or 2-by-2 pivot block will be used
425 *
426  absakk = abs( ap( kc ) )
427 *
428 * IMAX is the row-index of the largest off-diagonal element in
429 * column K, and COLMAX is its absolute value
430 *
431  IF( k.LT.n ) THEN
432  imax = k + isamax( n-k, ap( kc+1 ), 1 )
433  colmax = abs( ap( kc+imax-k ) )
434  ELSE
435  colmax = zero
436  END IF
437 *
438  IF( max( absakk, colmax ).EQ.zero ) THEN
439 *
440 * Column K is zero: set INFO and continue
441 *
442  IF( info.EQ.0 )
443  \$ info = k
444  kp = k
445  ELSE
446  IF( absakk.GE.alpha*colmax ) THEN
447 *
448 * no interchange, use 1-by-1 pivot block
449 *
450  kp = k
451  ELSE
452 *
453 * JMAX is the column-index of the largest off-diagonal
454 * element in row IMAX, and ROWMAX is its absolute value
455 *
456  rowmax = zero
457  kx = kc + imax - k
458  DO 70 j = k, imax - 1
459  IF( abs( ap( kx ) ).GT.rowmax ) THEN
460  rowmax = abs( ap( kx ) )
461  jmax = j
462  END IF
463  kx = kx + n - j
464  70 CONTINUE
465  kpc = npp - ( n-imax+1 )*( n-imax+2 ) / 2 + 1
466  IF( imax.LT.n ) THEN
467  jmax = imax + isamax( n-imax, ap( kpc+1 ), 1 )
468  rowmax = max( rowmax, abs( ap( kpc+jmax-imax ) ) )
469  END IF
470 *
471  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
472 *
473 * no interchange, use 1-by-1 pivot block
474 *
475  kp = k
476  ELSE IF( abs( ap( kpc ) ).GE.alpha*rowmax ) THEN
477 *
478 * interchange rows and columns K and IMAX, use 1-by-1
479 * pivot block
480 *
481  kp = imax
482  ELSE
483 *
484 * interchange rows and columns K+1 and IMAX, use 2-by-2
485 * pivot block
486 *
487  kp = imax
488  kstep = 2
489  END IF
490  END IF
491 *
492  kk = k + kstep - 1
493  IF( kstep.EQ.2 )
494  \$ knc = knc + n - k + 1
495  IF( kp.NE.kk ) THEN
496 *
497 * Interchange rows and columns KK and KP in the trailing
498 * submatrix A(k:n,k:n)
499 *
500  IF( kp.LT.n )
501  \$ CALL sswap( n-kp, ap( knc+kp-kk+1 ), 1, ap( kpc+1 ),
502  \$ 1 )
503  kx = knc + kp - kk
504  DO 80 j = kk + 1, kp - 1
505  kx = kx + n - j + 1
506  t = ap( knc+j-kk )
507  ap( knc+j-kk ) = ap( kx )
508  ap( kx ) = t
509  80 CONTINUE
510  t = ap( knc )
511  ap( knc ) = ap( kpc )
512  ap( kpc ) = t
513  IF( kstep.EQ.2 ) THEN
514  t = ap( kc+1 )
515  ap( kc+1 ) = ap( kc+kp-k )
516  ap( kc+kp-k ) = t
517  END IF
518  END IF
519 *
520 * Update the trailing submatrix
521 *
522  IF( kstep.EQ.1 ) THEN
523 *
524 * 1-by-1 pivot block D(k): column k now holds
525 *
526 * W(k) = L(k)*D(k)
527 *
528 * where L(k) is the k-th column of L
529 *
530  IF( k.LT.n ) THEN
531 *
532 * Perform a rank-1 update of A(k+1:n,k+1:n) as
533 *
534 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
535 *
536  r1 = one / ap( kc )
537  CALL sspr( uplo, n-k, -r1, ap( kc+1 ), 1,
538  \$ ap( kc+n-k+1 ) )
539 *
540 * Store L(k) in column K
541 *
542  CALL sscal( n-k, r1, ap( kc+1 ), 1 )
543  END IF
544  ELSE
545 *
546 * 2-by-2 pivot block D(k): columns K and K+1 now hold
547 *
548 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
549 *
550 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
551 * of L
552 *
553  IF( k.LT.n-1 ) THEN
554 *
555 * Perform a rank-2 update of A(k+2:n,k+2:n) as
556 *
557 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
558 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
559 *
560 * where L(k) and L(k+1) are the k-th and (k+1)-th
561 * columns of L
562 *
563  d21 = ap( k+1+( k-1 )*( 2*n-k ) / 2 )
564  d11 = ap( k+1+k*( 2*n-k-1 ) / 2 ) / d21
565  d22 = ap( k+( k-1 )*( 2*n-k ) / 2 ) / d21
566  t = one / ( d11*d22-one )
567  d21 = t / d21
568 *
569  DO 100 j = k + 2, n
570  wk = d21*( d11*ap( j+( k-1 )*( 2*n-k ) / 2 )-
571  \$ ap( j+k*( 2*n-k-1 ) / 2 ) )
572  wkp1 = d21*( d22*ap( j+k*( 2*n-k-1 ) / 2 )-
573  \$ ap( j+( k-1 )*( 2*n-k ) / 2 ) )
574 *
575  DO 90 i = j, n
576  ap( i+( j-1 )*( 2*n-j ) / 2 ) = ap( i+( j-1 )*
577  \$ ( 2*n-j ) / 2 ) - ap( i+( k-1 )*( 2*n-k ) /
578  \$ 2 )*wk - ap( i+k*( 2*n-k-1 ) / 2 )*wkp1
579  90 CONTINUE
580 *
581  ap( j+( k-1 )*( 2*n-k ) / 2 ) = wk
582  ap( j+k*( 2*n-k-1 ) / 2 ) = wkp1
583 *
584  100 CONTINUE
585  END IF
586  END IF
587  END IF
588 *
589 * Store details of the interchanges in IPIV
590 *
591  IF( kstep.EQ.1 ) THEN
592  ipiv( k ) = kp
593  ELSE
594  ipiv( k ) = -kp
595  ipiv( k+1 ) = -kp
596  END IF
597 *
598 * Increase K and return to the start of the main loop
599 *
600  k = k + kstep
601  kc = knc + n - k + 2
602  GO TO 60
603 *
604  END IF
605 *
606  110 CONTINUE
607  RETURN
608 *
609 * End of SSPTRF
610 *
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine sspr(UPLO, N, ALPHA, X, INCX, AP)
SSPR
Definition: sspr.f:127
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