LAPACK  3.9.1
LAPACK: Linear Algebra PACKage

◆ zgbtf2()

subroutine zgbtf2 ( integer  M,
integer  N,
integer  KL,
integer  KU,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
integer  INFO 
)

ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Download ZGBTF2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZGBTF2 computes an LU factorization of a complex m-by-n band matrix
 A using partial pivoting with row interchanges.

 This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrix A.  N >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in,out]AB
          AB is COMPLEX*16 array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[out]IPIV
          IPIV is INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).
[out]INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used
               to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

      *    *    *    +    +    +       *    *    *   u14  u25  u36
      *    *    +    +    +    +       *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked
  + need not be set on entry, but are required by the routine to store
  elements of U, because of fill-in resulting from the row
  interchanges.

Definition at line 144 of file zgbtf2.f.

145 *
146 * -- LAPACK computational routine --
147 * -- LAPACK is a software package provided by Univ. of Tennessee, --
148 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
149 *
150 * .. Scalar Arguments ..
151  INTEGER INFO, KL, KU, LDAB, M, N
152 * ..
153 * .. Array Arguments ..
154  INTEGER IPIV( * )
155  COMPLEX*16 AB( LDAB, * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  COMPLEX*16 ONE, ZERO
162  parameter( one = ( 1.0d+0, 0.0d+0 ),
163  $ zero = ( 0.0d+0, 0.0d+0 ) )
164 * ..
165 * .. Local Scalars ..
166  INTEGER I, J, JP, JU, KM, KV
167 * ..
168 * .. External Functions ..
169  INTEGER IZAMAX
170  EXTERNAL izamax
171 * ..
172 * .. External Subroutines ..
173  EXTERNAL xerbla, zgeru, zscal, zswap
174 * ..
175 * .. Intrinsic Functions ..
176  INTRINSIC max, min
177 * ..
178 * .. Executable Statements ..
179 *
180 * KV is the number of superdiagonals in the factor U, allowing for
181 * fill-in.
182 *
183  kv = ku + kl
184 *
185 * Test the input parameters.
186 *
187  info = 0
188  IF( m.LT.0 ) THEN
189  info = -1
190  ELSE IF( n.LT.0 ) THEN
191  info = -2
192  ELSE IF( kl.LT.0 ) THEN
193  info = -3
194  ELSE IF( ku.LT.0 ) THEN
195  info = -4
196  ELSE IF( ldab.LT.kl+kv+1 ) THEN
197  info = -6
198  END IF
199  IF( info.NE.0 ) THEN
200  CALL xerbla( 'ZGBTF2', -info )
201  RETURN
202  END IF
203 *
204 * Quick return if possible
205 *
206  IF( m.EQ.0 .OR. n.EQ.0 )
207  $ RETURN
208 *
209 * Gaussian elimination with partial pivoting
210 *
211 * Set fill-in elements in columns KU+2 to KV to zero.
212 *
213  DO 20 j = ku + 2, min( kv, n )
214  DO 10 i = kv - j + 2, kl
215  ab( i, j ) = zero
216  10 CONTINUE
217  20 CONTINUE
218 *
219 * JU is the index of the last column affected by the current stage
220 * of the factorization.
221 *
222  ju = 1
223 *
224  DO 40 j = 1, min( m, n )
225 *
226 * Set fill-in elements in column J+KV to zero.
227 *
228  IF( j+kv.LE.n ) THEN
229  DO 30 i = 1, kl
230  ab( i, j+kv ) = zero
231  30 CONTINUE
232  END IF
233 *
234 * Find pivot and test for singularity. KM is the number of
235 * subdiagonal elements in the current column.
236 *
237  km = min( kl, m-j )
238  jp = izamax( km+1, ab( kv+1, j ), 1 )
239  ipiv( j ) = jp + j - 1
240  IF( ab( kv+jp, j ).NE.zero ) THEN
241  ju = max( ju, min( j+ku+jp-1, n ) )
242 *
243 * Apply interchange to columns J to JU.
244 *
245  IF( jp.NE.1 )
246  $ CALL zswap( ju-j+1, ab( kv+jp, j ), ldab-1,
247  $ ab( kv+1, j ), ldab-1 )
248  IF( km.GT.0 ) THEN
249 *
250 * Compute multipliers.
251 *
252  CALL zscal( km, one / ab( kv+1, j ), ab( kv+2, j ), 1 )
253 *
254 * Update trailing submatrix within the band.
255 *
256  IF( ju.GT.j )
257  $ CALL zgeru( km, ju-j, -one, ab( kv+2, j ), 1,
258  $ ab( kv, j+1 ), ldab-1, ab( kv+1, j+1 ),
259  $ ldab-1 )
260  END IF
261  ELSE
262 *
263 * If pivot is zero, set INFO to the index of the pivot
264 * unless a zero pivot has already been found.
265 *
266  IF( info.EQ.0 )
267  $ info = j
268  END IF
269  40 CONTINUE
270  RETURN
271 *
272 * End of ZGBTF2
273 *
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zgeru(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZGERU
Definition: zgeru.f:130
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