LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dtbsv()

 subroutine dtbsv ( character UPLO, character TRANS, character DIAG, integer N, integer K, double precision, dimension(lda,*) A, integer LDA, double precision, dimension(*) X, integer INCX )

DTBSV

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

A*x = b,   or   A**T*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**T*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dtbsv.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  DOUBLE PRECISION A(LDA,*),X(*)
200 * ..
201 *
202 * =====================================================================
203 *
204 * .. Parameters ..
205  DOUBLE PRECISION ZERO
206  parameter(zero=0.0d+0)
207 * ..
208 * .. Local Scalars ..
209  DOUBLE PRECISION TEMP
210  INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
211  LOGICAL NOUNIT
212 * ..
213 * .. External Functions ..
214  LOGICAL LSAME
215  EXTERNAL lsame
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL xerbla
219 * ..
220 * .. Intrinsic Functions ..
221  INTRINSIC 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('DTBSV ',info)
245  RETURN
246  END IF
247 *
248 * Quick return if possible.
249 *
250  IF (n.EQ.0) RETURN
251 *
252  nounit = lsame(diag,'N')
253 *
254 * Set up the start point in X if the increment is not unity. This
255 * will be ( N - 1 )*INCX too small for descending loops.
256 *
257  IF (incx.LE.0) THEN
258  kx = 1 - (n-1)*incx
259  ELSE IF (incx.NE.1) THEN
260  kx = 1
261  END IF
262 *
263 * Start the operations. In this version the elements of A are
264 * accessed by sequentially with one pass through A.
265 *
266  IF (lsame(trans,'N')) THEN
267 *
268 * Form x := inv( A )*x.
269 *
270  IF (lsame(uplo,'U')) THEN
271  kplus1 = k + 1
272  IF (incx.EQ.1) THEN
273  DO 20 j = n,1,-1
274  IF (x(j).NE.zero) THEN
275  l = kplus1 - j
276  IF (nounit) x(j) = x(j)/a(kplus1,j)
277  temp = x(j)
278  DO 10 i = j - 1,max(1,j-k),-1
279  x(i) = x(i) - temp*a(l+i,j)
280  10 CONTINUE
281  END IF
282  20 CONTINUE
283  ELSE
284  kx = kx + (n-1)*incx
285  jx = kx
286  DO 40 j = n,1,-1
287  kx = kx - incx
288  IF (x(jx).NE.zero) THEN
289  ix = kx
290  l = kplus1 - j
291  IF (nounit) x(jx) = x(jx)/a(kplus1,j)
292  temp = x(jx)
293  DO 30 i = j - 1,max(1,j-k),-1
294  x(ix) = x(ix) - temp*a(l+i,j)
295  ix = ix - incx
296  30 CONTINUE
297  END IF
298  jx = jx - incx
299  40 CONTINUE
300  END IF
301  ELSE
302  IF (incx.EQ.1) THEN
303  DO 60 j = 1,n
304  IF (x(j).NE.zero) THEN
305  l = 1 - j
306  IF (nounit) x(j) = x(j)/a(1,j)
307  temp = x(j)
308  DO 50 i = j + 1,min(n,j+k)
309  x(i) = x(i) - temp*a(l+i,j)
310  50 CONTINUE
311  END IF
312  60 CONTINUE
313  ELSE
314  jx = kx
315  DO 80 j = 1,n
316  kx = kx + incx
317  IF (x(jx).NE.zero) THEN
318  ix = kx
319  l = 1 - j
320  IF (nounit) x(jx) = x(jx)/a(1,j)
321  temp = x(jx)
322  DO 70 i = j + 1,min(n,j+k)
323  x(ix) = x(ix) - temp*a(l+i,j)
324  ix = ix + incx
325  70 CONTINUE
326  END IF
327  jx = jx + incx
328  80 CONTINUE
329  END IF
330  END IF
331  ELSE
332 *
333 * Form x := inv( A**T)*x.
334 *
335  IF (lsame(uplo,'U')) THEN
336  kplus1 = k + 1
337  IF (incx.EQ.1) THEN
338  DO 100 j = 1,n
339  temp = x(j)
340  l = kplus1 - j
341  DO 90 i = max(1,j-k),j - 1
342  temp = temp - a(l+i,j)*x(i)
343  90 CONTINUE
344  IF (nounit) temp = temp/a(kplus1,j)
345  x(j) = temp
346  100 CONTINUE
347  ELSE
348  jx = kx
349  DO 120 j = 1,n
350  temp = x(jx)
351  ix = kx
352  l = kplus1 - j
353  DO 110 i = max(1,j-k),j - 1
354  temp = temp - a(l+i,j)*x(ix)
355  ix = ix + incx
356  110 CONTINUE
357  IF (nounit) temp = temp/a(kplus1,j)
358  x(jx) = temp
359  jx = jx + incx
360  IF (j.GT.k) kx = kx + incx
361  120 CONTINUE
362  END IF
363  ELSE
364  IF (incx.EQ.1) THEN
365  DO 140 j = n,1,-1
366  temp = x(j)
367  l = 1 - j
368  DO 130 i = min(n,j+k),j + 1,-1
369  temp = temp - a(l+i,j)*x(i)
370  130 CONTINUE
371  IF (nounit) temp = temp/a(1,j)
372  x(j) = temp
373  140 CONTINUE
374  ELSE
375  kx = kx + (n-1)*incx
376  jx = kx
377  DO 160 j = n,1,-1
378  temp = x(jx)
379  ix = kx
380  l = 1 - j
381  DO 150 i = min(n,j+k),j + 1,-1
382  temp = temp - a(l+i,j)*x(ix)
383  ix = ix - incx
384  150 CONTINUE
385  IF (nounit) temp = temp/a(1,j)
386  x(jx) = temp
387  jx = jx - incx
388  IF ((n-j).GE.k) kx = kx - incx
389  160 CONTINUE
390  END IF
391  END IF
392  END IF
393 *
394  RETURN
395 *
396 * End of DTBSV
397 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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