LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ ztftri()

subroutine ztftri ( character  transr,
character  uplo,
character  diag,
integer  n,
complex*16, dimension( 0: * )  a,
integer  info 
)

ZTFTRI

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

Purpose:
 ZTFTRI 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*16 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 ztftri.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*16 A( 0: * )
232* ..
233*
234* =====================================================================
235*
236* .. Parameters ..
237 COMPLEX*16 CONE
238 parameter( cone = ( 1.0d+0, 0.0d+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, ztrmm, ztrtri
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( 'ZTFTRI', -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 ztrtri( 'L', diag, n1, a( 0 ), n, info )
319 IF( info.GT.0 )
320 $ RETURN
321 CALL ztrmm( 'R', 'L', 'N', diag, n2, n1, -cone, a( 0 ),
322 $ n, a( n1 ), n )
323 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'L', diag, n1, a( n2 ), n, info )
338 IF( info.GT.0 )
339 $ RETURN
340 CALL ztrmm( 'L', 'L', 'C', diag, n1, n2, -cone, a( n2 ),
341 $ n, a( 0 ), n )
342 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'U', diag, n1, a( 0 ), n1, info )
362 IF( info.GT.0 )
363 $ RETURN
364 CALL ztrmm( 'L', 'U', 'N', diag, n1, n2, -cone, a( 0 ),
365 $ n1, a( n1*n1 ), n1 )
366 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'U', diag, n1, a( n2*n2 ), n2, info )
380 IF( info.GT.0 )
381 $ RETURN
382 CALL ztrmm( 'R', 'U', 'C', diag, n2, n1, -cone,
383 $ a( n2*n2 ), n2, a( 0 ), n2 )
384 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'L', diag, k, a( 1 ), n+1, info )
410 IF( info.GT.0 )
411 $ RETURN
412 CALL ztrmm( 'R', 'L', 'N', diag, k, k, -cone, a( 1 ),
413 $ n+1, a( k+1 ), n+1 )
414 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'L', diag, k, a( k+1 ), n+1, info )
429 IF( info.GT.0 )
430 $ RETURN
431 CALL ztrmm( 'L', 'L', 'C', diag, k, k, -cone, a( k+1 ),
432 $ n+1, a( 0 ), n+1 )
433 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'U', diag, k, a( k ), k, info )
452 IF( info.GT.0 )
453 $ RETURN
454 CALL ztrmm( 'L', 'U', 'N', diag, k, k, -cone, a( k ), k,
455 $ a( k*( k+1 ) ), k )
456 CALL ztrtri( '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 ztrmm( '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 ztrtri( 'U', diag, k, a( k*( k+1 ) ), k, info )
470 IF( info.GT.0 )
471 $ RETURN
472 CALL ztrmm( 'R', 'U', 'C', diag, k, k, -cone,
473 $ a( k*( k+1 ) ), k, a( 0 ), k )
474 CALL ztrtri( '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 ztrmm( '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 ZTFTRI
488*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
ztrmm
subroutine ztrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
ZTRMM
Definition ztrmm.f:177
ztrtri
subroutine ztrtri(uplo, diag, n, a, lda, info)
ZTRTRI
Definition ztrtri.f:109
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