1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
|
SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER ITRANS, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * )
* ..
*
* Purpose
* =======
*
* DGTTS2 solves one of the systems of equations
* A*X = B or A'*X = B,
* with a tridiagonal matrix A using the LU factorization computed
* by DGTTRF.
*
* Arguments
* =========
*
* ITRANS (input) INTEGER
* Specifies the form of the system of equations.
* = 0: A * X = B (No transpose)
* = 1: A'* X = B (Transpose)
* = 2: A'* X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* The order of the matrix A.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input) DOUBLE PRECISION array, dimension (N-1)
* The (n-1) multipliers that define the matrix L from the
* LU factorization of A.
*
* D (input) DOUBLE PRECISION array, dimension (N)
* The n diagonal elements of the upper triangular matrix U from
* the LU factorization of A.
*
* DU (input) DOUBLE PRECISION array, dimension (N-1)
* The (n-1) elements of the first super-diagonal of U.
*
* DU2 (input) DOUBLE PRECISION array, dimension (N-2)
* The (n-2) elements of the second super-diagonal of U.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the matrix of right hand side vectors B.
* On exit, B is overwritten by the solution vectors X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IP, J
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( ITRANS.EQ.0 ) THEN
*
* Solve A*X = B using the LU factorization of A,
* overwriting each right hand side vector with its solution.
*
IF( NRHS.LE.1 ) THEN
J = 1
10 CONTINUE
*
* Solve L*x = b.
*
DO 20 I = 1, N - 1
IP = IPIV( I )
TEMP = B( I+1-IP+I, J ) - DL( I )*B( IP, J )
B( I, J ) = B( IP, J )
B( I+1, J ) = TEMP
20 CONTINUE
*
* Solve U*x = b.
*
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
$ D( N-1 )
DO 30 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
$ B( I+2, J ) ) / D( I )
30 CONTINUE
IF( J.LT.NRHS ) THEN
J = J + 1
GO TO 10
END IF
ELSE
DO 60 J = 1, NRHS
*
* Solve L*x = b.
*
DO 40 I = 1, N - 1
IF( IPIV( I ).EQ.I ) THEN
B( I+1, J ) = B( I+1, J ) - DL( I )*B( I, J )
ELSE
TEMP = B( I, J )
B( I, J ) = B( I+1, J )
B( I+1, J ) = TEMP - DL( I )*B( I, J )
END IF
40 CONTINUE
*
* Solve U*x = b.
*
B( N, J ) = B( N, J ) / D( N )
IF( N.GT.1 )
$ B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) /
$ D( N-1 )
DO 50 I = N - 2, 1, -1
B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )*
$ B( I+2, J ) ) / D( I )
50 CONTINUE
60 CONTINUE
END IF
ELSE
*
* Solve A' * X = B.
*
IF( NRHS.LE.1 ) THEN
*
* Solve U'*x = b.
*
J = 1
70 CONTINUE
B( 1, J ) = B( 1, J ) / D( 1 )
IF( N.GT.1 )
$ B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
DO 80 I = 3, N
B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-DU2( I-2 )*
$ B( I-2, J ) ) / D( I )
80 CONTINUE
*
* Solve L'*x = b.
*
DO 90 I = N - 1, 1, -1
IP = IPIV( I )
TEMP = B( I, J ) - DL( I )*B( I+1, J )
B( I, J ) = B( IP, J )
B( IP, J ) = TEMP
90 CONTINUE
IF( J.LT.NRHS ) THEN
J = J + 1
GO TO 70
END IF
*
ELSE
DO 120 J = 1, NRHS
*
* Solve U'*x = b.
*
B( 1, J ) = B( 1, J ) / D( 1 )
IF( N.GT.1 )
$ B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 )
DO 100 I = 3, N
B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-
$ DU2( I-2 )*B( I-2, J ) ) / D( I )
100 CONTINUE
DO 110 I = N - 1, 1, -1
IF( IPIV( I ).EQ.I ) THEN
B( I, J ) = B( I, J ) - DL( I )*B( I+1, J )
ELSE
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - DL( I )*TEMP
B( I, J ) = TEMP
END IF
110 CONTINUE
120 CONTINUE
END IF
END IF
*
* End of DGTTS2
*
END
|