LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

◆ zpftri()

 subroutine zpftri ( character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, integer INFO )

ZPFTRI

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Purpose:
``` ZPFTRI computes the inverse of a complex Hermitian positive definite
matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
computed by ZPFTRF.```
Parameters
 [in] TRANSR ``` TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugate-transpose TRANSR of RFP A is stored.``` [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] A ``` A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the Hermitian matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the nt elements of upper packed A. If UPLO = 'L' the RFP A contains the elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N is odd. See the Note below for more details. On exit, the Hermitian inverse of the original matrix, in the same storage format.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed.```
Further Details:
```  We first consider Standard Packed Format when N is even.
We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N even and TRANSR = 'N'.

RFP A                   RFP A

-- -- --
03 04 05                33 43 53
-- --
13 14 15                00 44 54
--
23 24 25                10 11 55

33 34 35                20 21 22
--
00 44 45                30 31 32
-- --
01 11 55                40 41 42
-- -- --
02 12 22                50 51 52

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- -- --                -- -- -- -- -- --
03 13 23 33 00 01 02    33 00 10 20 30 40 50
-- -- -- -- --                -- -- -- -- --
04 14 24 34 44 11 12    43 44 11 21 31 41 51
-- -- -- -- -- --                -- -- -- --
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We next  consider Standard Packed Format when N is odd.
We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two   columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two   columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N odd  and TRANSR = 'N'.

RFP A                   RFP A

-- --
02 03 04                00 33 43
--
12 13 14                10 11 44

22 23 24                20 21 22
--
00 33 34                30 31 32
-- --
01 11 44                40 41 42

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- --                   -- -- -- -- -- --
02 12 22 00 01             00 10 20 30 40 50
-- -- -- --                   -- -- -- -- --
03 13 23 33 11             33 11 21 31 41 51
-- -- -- -- --                   -- -- -- --
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 211 of file zpftri.f.

212 *
213 * -- LAPACK computational routine --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 *
217 * .. Scalar Arguments ..
218  CHARACTER TRANSR, UPLO
219  INTEGER INFO, N
220 * .. Array Arguments ..
221  COMPLEX*16 A( 0: * )
222 * ..
223 *
224 * =====================================================================
225 *
226 * .. Parameters ..
227  DOUBLE PRECISION ONE
228  COMPLEX*16 CONE
229  parameter( one = 1.d0, cone = ( 1.d0, 0.d0 ) )
230 * ..
231 * .. Local Scalars ..
232  LOGICAL LOWER, NISODD, NORMALTRANSR
233  INTEGER N1, N2, K
234 * ..
235 * .. External Functions ..
236  LOGICAL LSAME
237  EXTERNAL lsame
238 * ..
239 * .. External Subroutines ..
240  EXTERNAL xerbla, ztftri, zlauum, ztrmm, zherk
241 * ..
242 * .. Intrinsic Functions ..
243  INTRINSIC mod
244 * ..
245 * .. Executable Statements ..
246 *
247 * Test the input parameters.
248 *
249  info = 0
250  normaltransr = lsame( transr, 'N' )
251  lower = lsame( uplo, 'L' )
252  IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'C' ) ) THEN
253  info = -1
254  ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
255  info = -2
256  ELSE IF( n.LT.0 ) THEN
257  info = -3
258  END IF
259  IF( info.NE.0 ) THEN
260  CALL xerbla( 'ZPFTRI', -info )
261  RETURN
262  END IF
263 *
264 * Quick return if possible
265 *
266  IF( n.EQ.0 )
267  \$ RETURN
268 *
269 * Invert the triangular Cholesky factor U or L.
270 *
271  CALL ztftri( transr, uplo, 'N', n, a, info )
272  IF( info.GT.0 )
273  \$ RETURN
274 *
275 * If N is odd, set NISODD = .TRUE.
276 * If N is even, set K = N/2 and NISODD = .FALSE.
277 *
278  IF( mod( n, 2 ).EQ.0 ) THEN
279  k = n / 2
280  nisodd = .false.
281  ELSE
282  nisodd = .true.
283  END IF
284 *
285 * Set N1 and N2 depending on LOWER
286 *
287  IF( lower ) THEN
288  n2 = n / 2
289  n1 = n - n2
290  ELSE
291  n1 = n / 2
292  n2 = n - n1
293  END IF
294 *
295 * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
296 * inv(L)^C*inv(L). There are eight cases.
297 *
298  IF( nisodd ) THEN
299 *
300 * N is odd
301 *
302  IF( normaltransr ) THEN
303 *
304 * N is odd and TRANSR = 'N'
305 *
306  IF( lower ) THEN
307 *
308 * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
309 * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
310 * T1 -> a(0), T2 -> a(n), S -> a(N1)
311 *
312  CALL zlauum( 'L', n1, a( 0 ), n, info )
313  CALL zherk( 'L', 'C', n1, n2, one, a( n1 ), n, one,
314  \$ a( 0 ), n )
315  CALL ztrmm( 'L', 'U', 'N', 'N', n2, n1, cone, a( n ), n,
316  \$ a( n1 ), n )
317  CALL zlauum( 'U', n2, a( n ), n, info )
318 *
319  ELSE
320 *
321 * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
322 * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
323 * T1 -> a(N2), T2 -> a(N1), S -> a(0)
324 *
325  CALL zlauum( 'L', n1, a( n2 ), n, info )
326  CALL zherk( 'L', 'N', n1, n2, one, a( 0 ), n, one,
327  \$ a( n2 ), n )
328  CALL ztrmm( 'R', 'U', 'C', 'N', n1, n2, cone, a( n1 ), n,
329  \$ a( 0 ), n )
330  CALL zlauum( 'U', n2, a( n1 ), n, info )
331 *
332  END IF
333 *
334  ELSE
335 *
336 * N is odd and TRANSR = 'C'
337 *
338  IF( lower ) THEN
339 *
340 * SRPA for LOWER, TRANSPOSE, and N is odd
341 * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
342 *
343  CALL zlauum( 'U', n1, a( 0 ), n1, info )
344  CALL zherk( 'U', 'N', n1, n2, one, a( n1*n1 ), n1, one,
345  \$ a( 0 ), n1 )
346  CALL ztrmm( 'R', 'L', 'N', 'N', n1, n2, cone, a( 1 ), n1,
347  \$ a( n1*n1 ), n1 )
348  CALL zlauum( 'L', n2, a( 1 ), n1, info )
349 *
350  ELSE
351 *
352 * SRPA for UPPER, TRANSPOSE, and N is odd
353 * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
354 *
355  CALL zlauum( 'U', n1, a( n2*n2 ), n2, info )
356  CALL zherk( 'U', 'C', n1, n2, one, a( 0 ), n2, one,
357  \$ a( n2*n2 ), n2 )
358  CALL ztrmm( 'L', 'L', 'C', 'N', n2, n1, cone, a( n1*n2 ),
359  \$ n2, a( 0 ), n2 )
360  CALL zlauum( 'L', n2, a( n1*n2 ), n2, info )
361 *
362  END IF
363 *
364  END IF
365 *
366  ELSE
367 *
368 * N is even
369 *
370  IF( normaltransr ) THEN
371 *
372 * N is even and TRANSR = 'N'
373 *
374  IF( lower ) THEN
375 *
376 * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
377 * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
378 * T1 -> a(1), T2 -> a(0), S -> a(k+1)
379 *
380  CALL zlauum( 'L', k, a( 1 ), n+1, info )
381  CALL zherk( 'L', 'C', k, k, one, a( k+1 ), n+1, one,
382  \$ a( 1 ), n+1 )
383  CALL ztrmm( 'L', 'U', 'N', 'N', k, k, cone, a( 0 ), n+1,
384  \$ a( k+1 ), n+1 )
385  CALL zlauum( 'U', k, a( 0 ), n+1, info )
386 *
387  ELSE
388 *
389 * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
390 * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
391 * T1 -> a(k+1), T2 -> a(k), S -> a(0)
392 *
393  CALL zlauum( 'L', k, a( k+1 ), n+1, info )
394  CALL zherk( 'L', 'N', k, k, one, a( 0 ), n+1, one,
395  \$ a( k+1 ), n+1 )
396  CALL ztrmm( 'R', 'U', 'C', 'N', k, k, cone, a( k ), n+1,
397  \$ a( 0 ), n+1 )
398  CALL zlauum( 'U', k, a( k ), n+1, info )
399 *
400  END IF
401 *
402  ELSE
403 *
404 * N is even and TRANSR = 'C'
405 *
406  IF( lower ) THEN
407 *
408 * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
409 * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
410 * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
411 *
412  CALL zlauum( 'U', k, a( k ), k, info )
413  CALL zherk( 'U', 'N', k, k, one, a( k*( k+1 ) ), k, one,
414  \$ a( k ), k )
415  CALL ztrmm( 'R', 'L', 'N', 'N', k, k, cone, a( 0 ), k,
416  \$ a( k*( k+1 ) ), k )
417  CALL zlauum( 'L', k, a( 0 ), k, info )
418 *
419  ELSE
420 *
421 * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
422 * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
423 * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
424 *
425  CALL zlauum( 'U', k, a( k*( k+1 ) ), k, info )
426  CALL zherk( 'U', 'C', k, k, one, a( 0 ), k, one,
427  \$ a( k*( k+1 ) ), k )
428  CALL ztrmm( 'L', 'L', 'C', 'N', k, k, cone, a( k*k ), k,
429  \$ a( 0 ), k )
430  CALL zlauum( 'L', k, a( k*k ), k, info )
431 *
432  END IF
433 *
434  END IF
435 *
436  END IF
437 *
438  RETURN
439 *
440 * End of ZPFTRI
441 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
ZHERK
Definition: zherk.f:173
subroutine ztrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRMM
Definition: ztrmm.f:177
subroutine zlauum(UPLO, N, A, LDA, INFO)
ZLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked...
Definition: zlauum.f:102
subroutine ztftri(TRANSR, UPLO, DIAG, N, A, INFO)
ZTFTRI
Definition: ztftri.f:221
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