LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
zgbtf2.f
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1 *> \brief \b ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGBTF2( 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 * COMPLEX*16 AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> ZGBTF2 computes an LU factorization of a complex m-by-n band matrix
38 *> A 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 COMPLEX*16 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 complex16GBcomputational
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 zgbtf2( 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  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 *
274  END
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
subroutine zgbtf2(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algo...
Definition: zgbtf2.f:145