1: *> \brief \b ZGBTF2
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: *> \date November 2011
119: *
120: *> \ingroup complex16GBcomputational
121: *
122: *> \par Further Details:
123: * =====================
124: *>
125: *> \verbatim
126: *>
127: *> The band storage scheme is illustrated by the following example, when
128: *> M = N = 6, KL = 2, KU = 1:
129: *>
130: *> On entry: On exit:
131: *>
132: *> * * * + + + * * * u14 u25 u36
133: *> * * + + + + * * u13 u24 u35 u46
134: *> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
135: *> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
136: *> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
137: *> a31 a42 a53 a64 * * m31 m42 m53 m64 * *
138: *>
139: *> Array elements marked * are not used by the routine; elements marked
140: *> + need not be set on entry, but are required by the routine to store
141: *> elements of U, because of fill-in resulting from the row
142: *> interchanges.
143: *> \endverbatim
144: *>
145: * =====================================================================
146: SUBROUTINE ZGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
147: *
148: * -- LAPACK computational routine (version 3.4.0) --
149: * -- LAPACK is a software package provided by Univ. of Tennessee, --
150: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
151: * November 2011
152: *
153: * .. Scalar Arguments ..
154: INTEGER INFO, KL, KU, LDAB, M, N
155: * ..
156: * .. Array Arguments ..
157: INTEGER IPIV( * )
158: COMPLEX*16 AB( LDAB, * )
159: * ..
160: *
161: * =====================================================================
162: *
163: * .. Parameters ..
164: COMPLEX*16 ONE, ZERO
165: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
166: $ ZERO = ( 0.0D+0, 0.0D+0 ) )
167: * ..
168: * .. Local Scalars ..
169: INTEGER I, J, JP, JU, KM, KV
170: * ..
171: * .. External Functions ..
172: INTEGER IZAMAX
173: EXTERNAL IZAMAX
174: * ..
175: * .. External Subroutines ..
176: EXTERNAL XERBLA, ZGERU, ZSCAL, ZSWAP
177: * ..
178: * .. Intrinsic Functions ..
179: INTRINSIC MAX, MIN
180: * ..
181: * .. Executable Statements ..
182: *
183: * KV is the number of superdiagonals in the factor U, allowing for
184: * fill-in.
185: *
186: KV = KU + KL
187: *
188: * Test the input parameters.
189: *
190: INFO = 0
191: IF( M.LT.0 ) THEN
192: INFO = -1
193: ELSE IF( N.LT.0 ) THEN
194: INFO = -2
195: ELSE IF( KL.LT.0 ) THEN
196: INFO = -3
197: ELSE IF( KU.LT.0 ) THEN
198: INFO = -4
199: ELSE IF( LDAB.LT.KL+KV+1 ) THEN
200: INFO = -6
201: END IF
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'ZGBTF2', -INFO )
204: RETURN
205: END IF
206: *
207: * Quick return if possible
208: *
209: IF( M.EQ.0 .OR. N.EQ.0 )
210: $ RETURN
211: *
212: * Gaussian elimination with partial pivoting
213: *
214: * Set fill-in elements in columns KU+2 to KV to zero.
215: *
216: DO 20 J = KU + 2, MIN( KV, N )
217: DO 10 I = KV - J + 2, KL
218: AB( I, J ) = ZERO
219: 10 CONTINUE
220: 20 CONTINUE
221: *
222: * JU is the index of the last column affected by the current stage
223: * of the factorization.
224: *
225: JU = 1
226: *
227: DO 40 J = 1, MIN( M, N )
228: *
229: * Set fill-in elements in column J+KV to zero.
230: *
231: IF( J+KV.LE.N ) THEN
232: DO 30 I = 1, KL
233: AB( I, J+KV ) = ZERO
234: 30 CONTINUE
235: END IF
236: *
237: * Find pivot and test for singularity. KM is the number of
238: * subdiagonal elements in the current column.
239: *
240: KM = MIN( KL, M-J )
241: JP = IZAMAX( KM+1, AB( KV+1, J ), 1 )
242: IPIV( J ) = JP + J - 1
243: IF( AB( KV+JP, J ).NE.ZERO ) THEN
244: JU = MAX( JU, MIN( J+KU+JP-1, N ) )
245: *
246: * Apply interchange to columns J to JU.
247: *
248: IF( JP.NE.1 )
249: $ CALL ZSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
250: $ AB( KV+1, J ), LDAB-1 )
251: IF( KM.GT.0 ) THEN
252: *
253: * Compute multipliers.
254: *
255: CALL ZSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ), 1 )
256: *
257: * Update trailing submatrix within the band.
258: *
259: IF( JU.GT.J )
260: $ CALL ZGERU( KM, JU-J, -ONE, AB( KV+2, J ), 1,
261: $ AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
262: $ LDAB-1 )
263: END IF
264: ELSE
265: *
266: * If pivot is zero, set INFO to the index of the pivot
267: * unless a zero pivot has already been found.
268: *
269: IF( INFO.EQ.0 )
270: $ INFO = J
271: END IF
272: 40 CONTINUE
273: RETURN
274: *
275: * End of ZGBTF2
276: *
277: END
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