LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ cpftri()

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

CPFTRI

Download CPFTRI + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CPFTRI 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 CPFTRF.
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 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 cpftri.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 A( 0: * )
222 * ..
223 *
224 * =====================================================================
225 *
226 * .. Parameters ..
227  REAL ONE
228  COMPLEX CONE
229  parameter( one = 1.0e+0, cone = ( 1.0e+0, 0.0e+0 ) )
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, ctftri, clauum, ctrmm, cherk
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( 'CPFTRI', -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 ctftri( 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 clauum( 'L', n1, a( 0 ), n, info )
313  CALL cherk( 'L', 'C', n1, n2, one, a( n1 ), n, one,
314  $ a( 0 ), n )
315  CALL ctrmm( 'L', 'U', 'N', 'N', n2, n1, cone, a( n ), n,
316  $ a( n1 ), n )
317  CALL clauum( '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 clauum( 'L', n1, a( n2 ), n, info )
326  CALL cherk( 'L', 'N', n1, n2, one, a( 0 ), n, one,
327  $ a( n2 ), n )
328  CALL ctrmm( 'R', 'U', 'C', 'N', n1, n2, cone, a( n1 ), n,
329  $ a( 0 ), n )
330  CALL clauum( '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 clauum( 'U', n1, a( 0 ), n1, info )
344  CALL cherk( 'U', 'N', n1, n2, one, a( n1*n1 ), n1, one,
345  $ a( 0 ), n1 )
346  CALL ctrmm( 'R', 'L', 'N', 'N', n1, n2, cone, a( 1 ), n1,
347  $ a( n1*n1 ), n1 )
348  CALL clauum( '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 clauum( 'U', n1, a( n2*n2 ), n2, info )
356  CALL cherk( 'U', 'C', n1, n2, one, a( 0 ), n2, one,
357  $ a( n2*n2 ), n2 )
358  CALL ctrmm( 'L', 'L', 'C', 'N', n2, n1, cone, a( n1*n2 ),
359  $ n2, a( 0 ), n2 )
360  CALL clauum( '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 clauum( 'L', k, a( 1 ), n+1, info )
381  CALL cherk( 'L', 'C', k, k, one, a( k+1 ), n+1, one,
382  $ a( 1 ), n+1 )
383  CALL ctrmm( 'L', 'U', 'N', 'N', k, k, cone, a( 0 ), n+1,
384  $ a( k+1 ), n+1 )
385  CALL clauum( '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 clauum( 'L', k, a( k+1 ), n+1, info )
394  CALL cherk( 'L', 'N', k, k, one, a( 0 ), n+1, one,
395  $ a( k+1 ), n+1 )
396  CALL ctrmm( 'R', 'U', 'C', 'N', k, k, cone, a( k ), n+1,
397  $ a( 0 ), n+1 )
398  CALL clauum( '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 clauum( 'U', k, a( k ), k, info )
413  CALL cherk( 'U', 'N', k, k, one, a( k*( k+1 ) ), k, one,
414  $ a( k ), k )
415  CALL ctrmm( 'R', 'L', 'N', 'N', k, k, cone, a( 0 ), k,
416  $ a( k*( k+1 ) ), k )
417  CALL clauum( '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 clauum( 'U', k, a( k*( k+1 ) ), k, info )
426  CALL cherk( 'U', 'C', k, k, one, a( 0 ), k, one,
427  $ a( k*( k+1 ) ), k )
428  CALL ctrmm( 'L', 'L', 'C', 'N', k, k, cone, a( k*k ), k,
429  $ a( 0 ), k )
430  CALL clauum( '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 CPFTRI
441 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ctrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM
Definition: ctrmm.f:177
subroutine cherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
CHERK
Definition: cherk.f:173
subroutine clauum(UPLO, N, A, LDA, INFO)
CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked...
Definition: clauum.f:102
subroutine ctftri(TRANSR, UPLO, DIAG, N, A, INFO)
CTFTRI
Definition: ctftri.f:221
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