 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ ctbsv()

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

CTBSV

Purpose:
``` CTBSV  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 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 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.```
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 188 of file ctbsv.f.

189 *
190 * -- Reference BLAS level2 routine --
191 * -- Reference BLAS is a software package provided by Univ. of Tennessee, --
192 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193 *
194 * .. Scalar Arguments ..
195  INTEGER INCX,K,LDA,N
196  CHARACTER DIAG,TRANS,UPLO
197 * ..
198 * .. Array Arguments ..
199  COMPLEX A(LDA,*),X(*)
200 * ..
201 *
202 * =====================================================================
203 *
204 * .. Parameters ..
205  COMPLEX ZERO
206  parameter(zero= (0.0e+0,0.0e+0))
207 * ..
208 * .. Local Scalars ..
209  COMPLEX TEMP
210  INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
211  LOGICAL NOCONJ,NOUNIT
212 * ..
213 * .. External Functions ..
214  LOGICAL LSAME
215  EXTERNAL lsame
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL xerbla
219 * ..
220 * .. Intrinsic Functions ..
221  INTRINSIC conjg,max,min
222 * ..
223 *
224 * Test the input parameters.
225 *
226  info = 0
227  IF (.NOT.lsame(uplo,'U') .AND. .NOT.lsame(uplo,'L')) THEN
228  info = 1
229  ELSE IF (.NOT.lsame(trans,'N') .AND. .NOT.lsame(trans,'T') .AND.
230  + .NOT.lsame(trans,'C')) THEN
231  info = 2
232  ELSE IF (.NOT.lsame(diag,'U') .AND. .NOT.lsame(diag,'N')) THEN
233  info = 3
234  ELSE IF (n.LT.0) THEN
235  info = 4
236  ELSE IF (k.LT.0) THEN
237  info = 5
238  ELSE IF (lda.LT. (k+1)) THEN
239  info = 7
240  ELSE IF (incx.EQ.0) THEN
241  info = 9
242  END IF
243  IF (info.NE.0) THEN
244  CALL xerbla('CTBSV ',info)
245  RETURN
246  END IF
247 *
248 * Quick return if possible.
249 *
250  IF (n.EQ.0) RETURN
251 *
252  noconj = lsame(trans,'T')
253  nounit = lsame(diag,'N')
254 *
255 * Set up the start point in X if the increment is not unity. This
256 * will be ( N - 1 )*INCX too small for descending loops.
257 *
258  IF (incx.LE.0) THEN
259  kx = 1 - (n-1)*incx
260  ELSE IF (incx.NE.1) THEN
261  kx = 1
262  END IF
263 *
264 * Start the operations. In this version the elements of A are
265 * accessed by sequentially with one pass through A.
266 *
267  IF (lsame(trans,'N')) THEN
268 *
269 * Form x := inv( A )*x.
270 *
271  IF (lsame(uplo,'U')) THEN
272  kplus1 = k + 1
273  IF (incx.EQ.1) THEN
274  DO 20 j = n,1,-1
275  IF (x(j).NE.zero) THEN
276  l = kplus1 - j
277  IF (nounit) x(j) = x(j)/a(kplus1,j)
278  temp = x(j)
279  DO 10 i = j - 1,max(1,j-k),-1
280  x(i) = x(i) - temp*a(l+i,j)
281  10 CONTINUE
282  END IF
283  20 CONTINUE
284  ELSE
285  kx = kx + (n-1)*incx
286  jx = kx
287  DO 40 j = n,1,-1
288  kx = kx - incx
289  IF (x(jx).NE.zero) THEN
290  ix = kx
291  l = kplus1 - j
292  IF (nounit) x(jx) = x(jx)/a(kplus1,j)
293  temp = x(jx)
294  DO 30 i = j - 1,max(1,j-k),-1
295  x(ix) = x(ix) - temp*a(l+i,j)
296  ix = ix - incx
297  30 CONTINUE
298  END IF
299  jx = jx - incx
300  40 CONTINUE
301  END IF
302  ELSE
303  IF (incx.EQ.1) THEN
304  DO 60 j = 1,n
305  IF (x(j).NE.zero) THEN
306  l = 1 - j
307  IF (nounit) x(j) = x(j)/a(1,j)
308  temp = x(j)
309  DO 50 i = j + 1,min(n,j+k)
310  x(i) = x(i) - temp*a(l+i,j)
311  50 CONTINUE
312  END IF
313  60 CONTINUE
314  ELSE
315  jx = kx
316  DO 80 j = 1,n
317  kx = kx + incx
318  IF (x(jx).NE.zero) THEN
319  ix = kx
320  l = 1 - j
321  IF (nounit) x(jx) = x(jx)/a(1,j)
322  temp = x(jx)
323  DO 70 i = j + 1,min(n,j+k)
324  x(ix) = x(ix) - temp*a(l+i,j)
325  ix = ix + incx
326  70 CONTINUE
327  END IF
328  jx = jx + incx
329  80 CONTINUE
330  END IF
331  END IF
332  ELSE
333 *
334 * Form x := inv( A**T )*x or x := inv( A**H )*x.
335 *
336  IF (lsame(uplo,'U')) THEN
337  kplus1 = k + 1
338  IF (incx.EQ.1) THEN
339  DO 110 j = 1,n
340  temp = x(j)
341  l = kplus1 - j
342  IF (noconj) THEN
343  DO 90 i = max(1,j-k),j - 1
344  temp = temp - a(l+i,j)*x(i)
345  90 CONTINUE
346  IF (nounit) temp = temp/a(kplus1,j)
347  ELSE
348  DO 100 i = max(1,j-k),j - 1
349  temp = temp - conjg(a(l+i,j))*x(i)
350  100 CONTINUE
351  IF (nounit) temp = temp/conjg(a(kplus1,j))
352  END IF
353  x(j) = temp
354  110 CONTINUE
355  ELSE
356  jx = kx
357  DO 140 j = 1,n
358  temp = x(jx)
359  ix = kx
360  l = kplus1 - j
361  IF (noconj) THEN
362  DO 120 i = max(1,j-k),j - 1
363  temp = temp - a(l+i,j)*x(ix)
364  ix = ix + incx
365  120 CONTINUE
366  IF (nounit) temp = temp/a(kplus1,j)
367  ELSE
368  DO 130 i = max(1,j-k),j - 1
369  temp = temp - conjg(a(l+i,j))*x(ix)
370  ix = ix + incx
371  130 CONTINUE
372  IF (nounit) temp = temp/conjg(a(kplus1,j))
373  END IF
374  x(jx) = temp
375  jx = jx + incx
376  IF (j.GT.k) kx = kx + incx
377  140 CONTINUE
378  END IF
379  ELSE
380  IF (incx.EQ.1) THEN
381  DO 170 j = n,1,-1
382  temp = x(j)
383  l = 1 - j
384  IF (noconj) THEN
385  DO 150 i = min(n,j+k),j + 1,-1
386  temp = temp - a(l+i,j)*x(i)
387  150 CONTINUE
388  IF (nounit) temp = temp/a(1,j)
389  ELSE
390  DO 160 i = min(n,j+k),j + 1,-1
391  temp = temp - conjg(a(l+i,j))*x(i)
392  160 CONTINUE
393  IF (nounit) temp = temp/conjg(a(1,j))
394  END IF
395  x(j) = temp
396  170 CONTINUE
397  ELSE
398  kx = kx + (n-1)*incx
399  jx = kx
400  DO 200 j = n,1,-1
401  temp = x(jx)
402  ix = kx
403  l = 1 - j
404  IF (noconj) THEN
405  DO 180 i = min(n,j+k),j + 1,-1
406  temp = temp - a(l+i,j)*x(ix)
407  ix = ix - incx
408  180 CONTINUE
409  IF (nounit) temp = temp/a(1,j)
410  ELSE
411  DO 190 i = min(n,j+k),j + 1,-1
412  temp = temp - conjg(a(l+i,j))*x(ix)
413  ix = ix - incx
414  190 CONTINUE
415  IF (nounit) temp = temp/conjg(a(1,j))
416  END IF
417  x(jx) = temp
418  jx = jx - incx
419  IF ((n-j).GE.k) kx = kx - incx
420  200 CONTINUE
421  END IF
422  END IF
423  END IF
424 *
425  RETURN
426 *
427 * End of CTBSV
428 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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