 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dgbtf2()

 subroutine dgbtf2 ( integer M, integer N, integer KL, integer KU, double precision, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, integer INFO )

DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose:
``` DGBTF2 computes an LU factorization of a real 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 DOUBLE PRECISION 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.```
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 dgbtf2.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  DOUBLE PRECISION AB( LDAB, * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  DOUBLE PRECISION ONE, ZERO
162  parameter( one = 1.0d+0, zero = 0.0d+0 )
163 * ..
164 * .. Local Scalars ..
165  INTEGER I, J, JP, JU, KM, KV
166 * ..
167 * .. External Functions ..
168  INTEGER IDAMAX
169  EXTERNAL idamax
170 * ..
171 * .. External Subroutines ..
172  EXTERNAL dger, dscal, dswap, xerbla
173 * ..
174 * .. Intrinsic Functions ..
175  INTRINSIC max, min
176 * ..
177 * .. Executable Statements ..
178 *
179 * KV is the number of superdiagonals in the factor U, allowing for
180 * fill-in.
181 *
182  kv = ku + kl
183 *
184 * Test the input parameters.
185 *
186  info = 0
187  IF( m.LT.0 ) THEN
188  info = -1
189  ELSE IF( n.LT.0 ) THEN
190  info = -2
191  ELSE IF( kl.LT.0 ) THEN
192  info = -3
193  ELSE IF( ku.LT.0 ) THEN
194  info = -4
195  ELSE IF( ldab.LT.kl+kv+1 ) THEN
196  info = -6
197  END IF
198  IF( info.NE.0 ) THEN
199  CALL xerbla( 'DGBTF2', -info )
200  RETURN
201  END IF
202 *
203 * Quick return if possible
204 *
205  IF( m.EQ.0 .OR. n.EQ.0 )
206  \$ RETURN
207 *
208 * Gaussian elimination with partial pivoting
209 *
210 * Set fill-in elements in columns KU+2 to KV to zero.
211 *
212  DO 20 j = ku + 2, min( kv, n )
213  DO 10 i = kv - j + 2, kl
214  ab( i, j ) = zero
215  10 CONTINUE
216  20 CONTINUE
217 *
218 * JU is the index of the last column affected by the current stage
219 * of the factorization.
220 *
221  ju = 1
222 *
223  DO 40 j = 1, min( m, n )
224 *
225 * Set fill-in elements in column J+KV to zero.
226 *
227  IF( j+kv.LE.n ) THEN
228  DO 30 i = 1, kl
229  ab( i, j+kv ) = zero
230  30 CONTINUE
231  END IF
232 *
233 * Find pivot and test for singularity. KM is the number of
234 * subdiagonal elements in the current column.
235 *
236  km = min( kl, m-j )
237  jp = idamax( km+1, ab( kv+1, j ), 1 )
238  ipiv( j ) = jp + j - 1
239  IF( ab( kv+jp, j ).NE.zero ) THEN
240  ju = max( ju, min( j+ku+jp-1, n ) )
241 *
242 * Apply interchange to columns J to JU.
243 *
244  IF( jp.NE.1 )
245  \$ CALL dswap( ju-j+1, ab( kv+jp, j ), ldab-1,
246  \$ ab( kv+1, j ), ldab-1 )
247 *
248  IF( km.GT.0 ) THEN
249 *
250 * Compute multipliers.
251 *
252  CALL dscal( 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 dger( 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 DGBTF2
273 *
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
DGER
Definition: dger.f:130
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