1: *> \brief \b ZGBTF2 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
9: *> Download ZGBTF2 + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbtf2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
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
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