LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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sgbtf2.f
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1*> \brief \b SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgbtf2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, KL, KU, LDAB, M, N
25* ..
26* .. Array Arguments ..
27* INTEGER IPIV( * )
28* REAL AB( LDAB, * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> SGBTF2 computes an LU factorization of a real m-by-n band matrix A
38*> using partial pivoting with row interchanges.
39*>
40*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
41*> \endverbatim
42*
43* Arguments:
44* ==========
45*
46*> \param[in] M
47*> \verbatim
48*> M is INTEGER
49*> The number of rows of the matrix A. M >= 0.
50*> \endverbatim
51*>
52*> \param[in] N
53*> \verbatim
54*> N is INTEGER
55*> The number of columns of the matrix A. N >= 0.
56*> \endverbatim
57*>
58*> \param[in] KL
59*> \verbatim
60*> KL is INTEGER
61*> The number of subdiagonals within the band of A. KL >= 0.
62*> \endverbatim
63*>
64*> \param[in] KU
65*> \verbatim
66*> KU is INTEGER
67*> The number of superdiagonals within the band of A. KU >= 0.
68*> \endverbatim
69*>
70*> \param[in,out] AB
71*> \verbatim
72*> AB is REAL array, dimension (LDAB,N)
73*> On entry, the matrix A in band storage, in rows KL+1 to
74*> 2*KL+KU+1; rows 1 to KL of the array need not be set.
75*> The j-th column of A is stored in the j-th column of the
76*> array AB as follows:
77*> AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
78*>
79*> On exit, details of the factorization: U is stored as an
80*> upper triangular band matrix with KL+KU superdiagonals in
81*> rows 1 to KL+KU+1, and the multipliers used during the
82*> factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
83*> See below for further details.
84*> \endverbatim
85*>
86*> \param[in] LDAB
87*> \verbatim
88*> LDAB is INTEGER
89*> The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
90*> \endverbatim
91*>
92*> \param[out] IPIV
93*> \verbatim
94*> IPIV is INTEGER array, dimension (min(M,N))
95*> The pivot indices; for 1 <= i <= min(M,N), row i of the
96*> matrix was interchanged with row IPIV(i).
97*> \endverbatim
98*>
99*> \param[out] INFO
100*> \verbatim
101*> INFO is INTEGER
102*> = 0: successful exit
103*> < 0: if INFO = -i, the i-th argument had an illegal value
104*> > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
105*> has been completed, but the factor U is exactly
106*> singular, and division by zero will occur if it is used
107*> to solve a system of equations.
108*> \endverbatim
109*
110* Authors:
111* ========
112*
113*> \author Univ. of Tennessee
114*> \author Univ. of California Berkeley
115*> \author Univ. of Colorado Denver
116*> \author NAG Ltd.
117*
118*> \ingroup realGBcomputational
119*
120*> \par Further Details:
121* =====================
122*>
123*> \verbatim
124*>
125*> The band storage scheme is illustrated by the following example, when
126*> M = N = 6, KL = 2, KU = 1:
127*>
128*> On entry: On exit:
129*>
130*> * * * + + + * * * u14 u25 u36
131*> * * + + + + * * u13 u24 u35 u46
132*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
133*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
134*> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
135*> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
136*>
137*> Array elements marked * are not used by the routine; elements marked
138*> + need not be set on entry, but are required by the routine to store
139*> elements of U, because of fill-in resulting from the row
140*> interchanges.
141*> \endverbatim
142*>
143* =====================================================================
144 SUBROUTINE sgbtf2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
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 REAL AB( LDAB, * )
156* ..
157*
158* =====================================================================
159*
160* .. Parameters ..
161 REAL ONE, ZERO
162 parameter( one = 1.0e+0, zero = 0.0e+0 )
163* ..
164* .. Local Scalars ..
165 INTEGER I, J, JP, JU, KM, KV
166* ..
167* .. External Functions ..
168 INTEGER ISAMAX
169 EXTERNAL isamax
170* ..
171* .. External Subroutines ..
172 EXTERNAL sger, sscal, sswap, 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( 'SGBTF2', -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 = isamax( 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 sswap( 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 sscal( 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 sger( 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 SGBTF2
273*
274 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgbtf2(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algo...
Definition: sgbtf2.f:145
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine sger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SGER
Definition: sger.f:130