LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ ctftri()

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

CTFTRI

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Purpose:
 CTFTRI computes the inverse of a triangular matrix A stored in RFP
 format.

 This is a Level 3 BLAS version of the algorithm.
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':  A is upper triangular;
          = 'L':  A is lower triangular.
[in]DIAG
          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.
[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 triangular 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 nt
          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 (triangular) 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, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can 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 220 of file ctftri.f.

221 *
222 * -- LAPACK computational routine --
223 * -- LAPACK is a software package provided by Univ. of Tennessee, --
224 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
225 *
226 * .. Scalar Arguments ..
227  CHARACTER TRANSR, UPLO, DIAG
228  INTEGER INFO, N
229 * ..
230 * .. Array Arguments ..
231  COMPLEX A( 0: * )
232 * ..
233 *
234 * =====================================================================
235 *
236 * .. Parameters ..
237  COMPLEX CONE
238  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
239 * ..
240 * .. Local Scalars ..
241  LOGICAL LOWER, NISODD, NORMALTRANSR
242  INTEGER N1, N2, K
243 * ..
244 * .. External Functions ..
245  LOGICAL LSAME
246  EXTERNAL lsame
247 * ..
248 * .. External Subroutines ..
249  EXTERNAL xerbla, ctrmm, ctrtri
250 * ..
251 * .. Intrinsic Functions ..
252  INTRINSIC mod
253 * ..
254 * .. Executable Statements ..
255 *
256 * Test the input parameters.
257 *
258  info = 0
259  normaltransr = lsame( transr, 'N' )
260  lower = lsame( uplo, 'L' )
261  IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'C' ) ) THEN
262  info = -1
263  ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
264  info = -2
265  ELSE IF( .NOT.lsame( diag, 'N' ) .AND. .NOT.lsame( diag, 'U' ) )
266  $ THEN
267  info = -3
268  ELSE IF( n.LT.0 ) THEN
269  info = -4
270  END IF
271  IF( info.NE.0 ) THEN
272  CALL xerbla( 'CTFTRI', -info )
273  RETURN
274  END IF
275 *
276 * Quick return if possible
277 *
278  IF( n.EQ.0 )
279  $ RETURN
280 *
281 * If N is odd, set NISODD = .TRUE.
282 * If N is even, set K = N/2 and NISODD = .FALSE.
283 *
284  IF( mod( n, 2 ).EQ.0 ) THEN
285  k = n / 2
286  nisodd = .false.
287  ELSE
288  nisodd = .true.
289  END IF
290 *
291 * Set N1 and N2 depending on LOWER
292 *
293  IF( lower ) THEN
294  n2 = n / 2
295  n1 = n - n2
296  ELSE
297  n1 = n / 2
298  n2 = n - n1
299  END IF
300 *
301 *
302 * start execution: there are eight cases
303 *
304  IF( nisodd ) THEN
305 *
306 * N is odd
307 *
308  IF( normaltransr ) THEN
309 *
310 * N is odd and TRANSR = 'N'
311 *
312  IF( lower ) THEN
313 *
314 * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
315 * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
316 * T1 -> a(0), T2 -> a(n), S -> a(n1)
317 *
318  CALL ctrtri( 'L', diag, n1, a( 0 ), n, info )
319  IF( info.GT.0 )
320  $ RETURN
321  CALL ctrmm( 'R', 'L', 'N', diag, n2, n1, -cone, a( 0 ),
322  $ n, a( n1 ), n )
323  CALL ctrtri( 'U', diag, n2, a( n ), n, info )
324  IF( info.GT.0 )
325  $ info = info + n1
326  IF( info.GT.0 )
327  $ RETURN
328  CALL ctrmm( 'L', 'U', 'C', diag, n2, n1, cone, a( n ), n,
329  $ a( n1 ), n )
330 *
331  ELSE
332 *
333 * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
334 * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
335 * T1 -> a(n2), T2 -> a(n1), S -> a(0)
336 *
337  CALL ctrtri( 'L', diag, n1, a( n2 ), n, info )
338  IF( info.GT.0 )
339  $ RETURN
340  CALL ctrmm( 'L', 'L', 'C', diag, n1, n2, -cone, a( n2 ),
341  $ n, a( 0 ), n )
342  CALL ctrtri( 'U', diag, n2, a( n1 ), n, info )
343  IF( info.GT.0 )
344  $ info = info + n1
345  IF( info.GT.0 )
346  $ RETURN
347  CALL ctrmm( 'R', 'U', 'N', diag, n1, n2, cone, a( n1 ),
348  $ n, a( 0 ), n )
349 *
350  END IF
351 *
352  ELSE
353 *
354 * N is odd and TRANSR = 'C'
355 *
356  IF( lower ) THEN
357 *
358 * SRPA for LOWER, TRANSPOSE and N is odd
359 * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
360 *
361  CALL ctrtri( 'U', diag, n1, a( 0 ), n1, info )
362  IF( info.GT.0 )
363  $ RETURN
364  CALL ctrmm( 'L', 'U', 'N', diag, n1, n2, -cone, a( 0 ),
365  $ n1, a( n1*n1 ), n1 )
366  CALL ctrtri( 'L', diag, n2, a( 1 ), n1, info )
367  IF( info.GT.0 )
368  $ info = info + n1
369  IF( info.GT.0 )
370  $ RETURN
371  CALL ctrmm( 'R', 'L', 'C', diag, n1, n2, cone, a( 1 ),
372  $ n1, a( n1*n1 ), n1 )
373 *
374  ELSE
375 *
376 * SRPA for UPPER, TRANSPOSE and N is odd
377 * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
378 *
379  CALL ctrtri( 'U', diag, n1, a( n2*n2 ), n2, info )
380  IF( info.GT.0 )
381  $ RETURN
382  CALL ctrmm( 'R', 'U', 'C', diag, n2, n1, -cone,
383  $ a( n2*n2 ), n2, a( 0 ), n2 )
384  CALL ctrtri( 'L', diag, n2, a( n1*n2 ), n2, info )
385  IF( info.GT.0 )
386  $ info = info + n1
387  IF( info.GT.0 )
388  $ RETURN
389  CALL ctrmm( 'L', 'L', 'N', diag, n2, n1, cone,
390  $ a( n1*n2 ), n2, a( 0 ), n2 )
391  END IF
392 *
393  END IF
394 *
395  ELSE
396 *
397 * N is even
398 *
399  IF( normaltransr ) THEN
400 *
401 * N is even and TRANSR = 'N'
402 *
403  IF( lower ) THEN
404 *
405 * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
406 * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
407 * T1 -> a(1), T2 -> a(0), S -> a(k+1)
408 *
409  CALL ctrtri( 'L', diag, k, a( 1 ), n+1, info )
410  IF( info.GT.0 )
411  $ RETURN
412  CALL ctrmm( 'R', 'L', 'N', diag, k, k, -cone, a( 1 ),
413  $ n+1, a( k+1 ), n+1 )
414  CALL ctrtri( 'U', diag, k, a( 0 ), n+1, info )
415  IF( info.GT.0 )
416  $ info = info + k
417  IF( info.GT.0 )
418  $ RETURN
419  CALL ctrmm( 'L', 'U', 'C', diag, k, k, cone, a( 0 ), n+1,
420  $ a( k+1 ), n+1 )
421 *
422  ELSE
423 *
424 * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
425 * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
426 * T1 -> a(k+1), T2 -> a(k), S -> a(0)
427 *
428  CALL ctrtri( 'L', diag, k, a( k+1 ), n+1, info )
429  IF( info.GT.0 )
430  $ RETURN
431  CALL ctrmm( 'L', 'L', 'C', diag, k, k, -cone, a( k+1 ),
432  $ n+1, a( 0 ), n+1 )
433  CALL ctrtri( 'U', diag, k, a( k ), n+1, info )
434  IF( info.GT.0 )
435  $ info = info + k
436  IF( info.GT.0 )
437  $ RETURN
438  CALL ctrmm( 'R', 'U', 'N', diag, k, k, cone, a( k ), n+1,
439  $ a( 0 ), n+1 )
440  END IF
441  ELSE
442 *
443 * N is even and TRANSR = 'C'
444 *
445  IF( lower ) THEN
446 *
447 * SRPA for LOWER, TRANSPOSE and N is even (see paper)
448 * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
449 * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
450 *
451  CALL ctrtri( 'U', diag, k, a( k ), k, info )
452  IF( info.GT.0 )
453  $ RETURN
454  CALL ctrmm( 'L', 'U', 'N', diag, k, k, -cone, a( k ), k,
455  $ a( k*( k+1 ) ), k )
456  CALL ctrtri( 'L', diag, k, a( 0 ), k, info )
457  IF( info.GT.0 )
458  $ info = info + k
459  IF( info.GT.0 )
460  $ RETURN
461  CALL ctrmm( 'R', 'L', 'C', diag, k, k, cone, a( 0 ), k,
462  $ a( k*( k+1 ) ), k )
463  ELSE
464 *
465 * SRPA for UPPER, TRANSPOSE and N is even (see paper)
466 * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
467 * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
468 *
469  CALL ctrtri( 'U', diag, k, a( k*( k+1 ) ), k, info )
470  IF( info.GT.0 )
471  $ RETURN
472  CALL ctrmm( 'R', 'U', 'C', diag, k, k, -cone,
473  $ a( k*( k+1 ) ), k, a( 0 ), k )
474  CALL ctrtri( 'L', diag, k, a( k*k ), k, info )
475  IF( info.GT.0 )
476  $ info = info + k
477  IF( info.GT.0 )
478  $ RETURN
479  CALL ctrmm( 'L', 'L', 'N', diag, k, k, cone, a( k*k ), k,
480  $ a( 0 ), k )
481  END IF
482  END IF
483  END IF
484 *
485  RETURN
486 *
487 * End of CTFTRI
488 *
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 ctrtri(UPLO, DIAG, N, A, LDA, INFO)
CTRTRI
Definition: ctrtri.f:109
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