 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztbsv()

 subroutine ztbsv ( character UPLO, character TRANS, character DIAG, integer N, integer K, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX )

ZTBSV

Purpose:
``` ZTBSV  solves one of the systems of equations

A*x = b,   or   A**T*x = b,   or   A**H*x = b,

where b and x are n element vectors and A is an n by n unit, or
non-unit, upper or lower triangular band matrix, with ( k + 1 )
diagonals.

No test for singularity or near-singularity is included in this
routine. Such tests must be performed before calling this routine.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix.``` [in] TRANS ``` TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**H*x = b.``` [in] DIAG ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular.``` [in] N ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.``` [in] K ``` K is INTEGER On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K.``` [in] A ``` A is COMPLEX*16 array, dimension ( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( k + 1 ) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced. The following program segment will transfer an upper triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced. The following program segment will transfer a lower triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.``` [in] LDA ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ).``` [in,out] X ``` X is COMPLEX*16 array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x.``` [in] INCX ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.```
Date
December 2016
Further Details:
```  Level 2 Blas routine.

-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.```

Definition at line 191 of file ztbsv.f.

191 *
192 * -- Reference BLAS level2 routine (version 3.7.0) --
193 * -- Reference BLAS is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 * December 2016
196 *
197 * .. Scalar Arguments ..
198  INTEGER incx,k,lda,n
199  CHARACTER diag,trans,uplo
200 * ..
201 * .. Array Arguments ..
202  COMPLEX*16 a(lda,*),x(*)
203 * ..
204 *
205 * =====================================================================
206 *
207 * .. Parameters ..
208  COMPLEX*16 zero
209  parameter(zero= (0.0d+0,0.0d+0))
210 * ..
211 * .. Local Scalars ..
212  COMPLEX*16 temp
213  INTEGER i,info,ix,j,jx,kplus1,kx,l
214  LOGICAL noconj,nounit
215 * ..
216 * .. External Functions ..
217  LOGICAL lsame
218  EXTERNAL lsame
219 * ..
220 * .. External Subroutines ..
221  EXTERNAL xerbla
222 * ..
223 * .. Intrinsic Functions ..
224  INTRINSIC dconjg,max,min
225 * ..
226 *
227 * Test the input parameters.
228 *
229  info = 0
230  IF (.NOT.lsame(uplo,'U') .AND. .NOT.lsame(uplo,'L')) THEN
231  info = 1
232  ELSE IF (.NOT.lsame(trans,'N') .AND. .NOT.lsame(trans,'T') .AND.
233  + .NOT.lsame(trans,'C')) THEN
234  info = 2
235  ELSE IF (.NOT.lsame(diag,'U') .AND. .NOT.lsame(diag,'N')) THEN
236  info = 3
237  ELSE IF (n.LT.0) THEN
238  info = 4
239  ELSE IF (k.LT.0) THEN
240  info = 5
241  ELSE IF (lda.LT. (k+1)) THEN
242  info = 7
243  ELSE IF (incx.EQ.0) THEN
244  info = 9
245  END IF
246  IF (info.NE.0) THEN
247  CALL xerbla('ZTBSV ',info)
248  RETURN
249  END IF
250 *
251 * Quick return if possible.
252 *
253  IF (n.EQ.0) RETURN
254 *
255  noconj = lsame(trans,'T')
256  nounit = lsame(diag,'N')
257 *
258 * Set up the start point in X if the increment is not unity. This
259 * will be ( N - 1 )*INCX too small for descending loops.
260 *
261  IF (incx.LE.0) THEN
262  kx = 1 - (n-1)*incx
263  ELSE IF (incx.NE.1) THEN
264  kx = 1
265  END IF
266 *
267 * Start the operations. In this version the elements of A are
268 * accessed by sequentially with one pass through A.
269 *
270  IF (lsame(trans,'N')) THEN
271 *
272 * Form x := inv( A )*x.
273 *
274  IF (lsame(uplo,'U')) THEN
275  kplus1 = k + 1
276  IF (incx.EQ.1) THEN
277  DO 20 j = n,1,-1
278  IF (x(j).NE.zero) THEN
279  l = kplus1 - j
280  IF (nounit) x(j) = x(j)/a(kplus1,j)
281  temp = x(j)
282  DO 10 i = j - 1,max(1,j-k),-1
283  x(i) = x(i) - temp*a(l+i,j)
284  10 CONTINUE
285  END IF
286  20 CONTINUE
287  ELSE
288  kx = kx + (n-1)*incx
289  jx = kx
290  DO 40 j = n,1,-1
291  kx = kx - incx
292  IF (x(jx).NE.zero) THEN
293  ix = kx
294  l = kplus1 - j
295  IF (nounit) x(jx) = x(jx)/a(kplus1,j)
296  temp = x(jx)
297  DO 30 i = j - 1,max(1,j-k),-1
298  x(ix) = x(ix) - temp*a(l+i,j)
299  ix = ix - incx
300  30 CONTINUE
301  END IF
302  jx = jx - incx
303  40 CONTINUE
304  END IF
305  ELSE
306  IF (incx.EQ.1) THEN
307  DO 60 j = 1,n
308  IF (x(j).NE.zero) THEN
309  l = 1 - j
310  IF (nounit) x(j) = x(j)/a(1,j)
311  temp = x(j)
312  DO 50 i = j + 1,min(n,j+k)
313  x(i) = x(i) - temp*a(l+i,j)
314  50 CONTINUE
315  END IF
316  60 CONTINUE
317  ELSE
318  jx = kx
319  DO 80 j = 1,n
320  kx = kx + incx
321  IF (x(jx).NE.zero) THEN
322  ix = kx
323  l = 1 - j
324  IF (nounit) x(jx) = x(jx)/a(1,j)
325  temp = x(jx)
326  DO 70 i = j + 1,min(n,j+k)
327  x(ix) = x(ix) - temp*a(l+i,j)
328  ix = ix + incx
329  70 CONTINUE
330  END IF
331  jx = jx + incx
332  80 CONTINUE
333  END IF
334  END IF
335  ELSE
336 *
337 * Form x := inv( A**T )*x or x := inv( A**H )*x.
338 *
339  IF (lsame(uplo,'U')) THEN
340  kplus1 = k + 1
341  IF (incx.EQ.1) THEN
342  DO 110 j = 1,n
343  temp = x(j)
344  l = kplus1 - j
345  IF (noconj) THEN
346  DO 90 i = max(1,j-k),j - 1
347  temp = temp - a(l+i,j)*x(i)
348  90 CONTINUE
349  IF (nounit) temp = temp/a(kplus1,j)
350  ELSE
351  DO 100 i = max(1,j-k),j - 1
352  temp = temp - dconjg(a(l+i,j))*x(i)
353  100 CONTINUE
354  IF (nounit) temp = temp/dconjg(a(kplus1,j))
355  END IF
356  x(j) = temp
357  110 CONTINUE
358  ELSE
359  jx = kx
360  DO 140 j = 1,n
361  temp = x(jx)
362  ix = kx
363  l = kplus1 - j
364  IF (noconj) THEN
365  DO 120 i = max(1,j-k),j - 1
366  temp = temp - a(l+i,j)*x(ix)
367  ix = ix + incx
368  120 CONTINUE
369  IF (nounit) temp = temp/a(kplus1,j)
370  ELSE
371  DO 130 i = max(1,j-k),j - 1
372  temp = temp - dconjg(a(l+i,j))*x(ix)
373  ix = ix + incx
374  130 CONTINUE
375  IF (nounit) temp = temp/dconjg(a(kplus1,j))
376  END IF
377  x(jx) = temp
378  jx = jx + incx
379  IF (j.GT.k) kx = kx + incx
380  140 CONTINUE
381  END IF
382  ELSE
383  IF (incx.EQ.1) THEN
384  DO 170 j = n,1,-1
385  temp = x(j)
386  l = 1 - j
387  IF (noconj) THEN
388  DO 150 i = min(n,j+k),j + 1,-1
389  temp = temp - a(l+i,j)*x(i)
390  150 CONTINUE
391  IF (nounit) temp = temp/a(1,j)
392  ELSE
393  DO 160 i = min(n,j+k),j + 1,-1
394  temp = temp - dconjg(a(l+i,j))*x(i)
395  160 CONTINUE
396  IF (nounit) temp = temp/dconjg(a(1,j))
397  END IF
398  x(j) = temp
399  170 CONTINUE
400  ELSE
401  kx = kx + (n-1)*incx
402  jx = kx
403  DO 200 j = n,1,-1
404  temp = x(jx)
405  ix = kx
406  l = 1 - j
407  IF (noconj) THEN
408  DO 180 i = min(n,j+k),j + 1,-1
409  temp = temp - a(l+i,j)*x(ix)
410  ix = ix - incx
411  180 CONTINUE
412  IF (nounit) temp = temp/a(1,j)
413  ELSE
414  DO 190 i = min(n,j+k),j + 1,-1
415  temp = temp - dconjg(a(l+i,j))*x(ix)
416  ix = ix - incx
417  190 CONTINUE
418  IF (nounit) temp = temp/dconjg(a(1,j))
419  END IF
420  x(jx) = temp
421  jx = jx - incx
422  IF ((n-j).GE.k) kx = kx - incx
423  200 CONTINUE
424  END IF
425  END IF
426  END IF
427 *
428  RETURN
429 *
430 * End of ZTBSV .
431 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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