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sptcon.f
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1 *> \brief \b SPTCON
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SPTCON + dependencies
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, N
25 * REAL ANORM, RCOND
26 * ..
27 * .. Array Arguments ..
28 * REAL D( * ), E( * ), WORK( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SPTCON computes the reciprocal of the condition number (in the
38 *> 1-norm) of a real symmetric positive definite tridiagonal matrix
39 *> using the factorization A = L*D*L**T or A = U**T*D*U computed by
40 *> SPTTRF.
41 *>
42 *> Norm(inv(A)) is computed by a direct method, and the reciprocal of
43 *> the condition number is computed as
44 *> RCOND = 1 / (ANORM * norm(inv(A))).
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] N
51 *> \verbatim
52 *> N is INTEGER
53 *> The order of the matrix A. N >= 0.
54 *> \endverbatim
55 *>
56 *> \param[in] D
57 *> \verbatim
58 *> D is REAL array, dimension (N)
59 *> The n diagonal elements of the diagonal matrix D from the
60 *> factorization of A, as computed by SPTTRF.
61 *> \endverbatim
62 *>
63 *> \param[in] E
64 *> \verbatim
65 *> E is REAL array, dimension (N-1)
66 *> The (n-1) off-diagonal elements of the unit bidiagonal factor
67 *> U or L from the factorization of A, as computed by SPTTRF.
68 *> \endverbatim
69 *>
70 *> \param[in] ANORM
71 *> \verbatim
72 *> ANORM is REAL
73 *> The 1-norm of the original matrix A.
74 *> \endverbatim
75 *>
76 *> \param[out] RCOND
77 *> \verbatim
78 *> RCOND is REAL
79 *> The reciprocal of the condition number of the matrix A,
80 *> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
81 *> 1-norm of inv(A) computed in this routine.
82 *> \endverbatim
83 *>
84 *> \param[out] WORK
85 *> \verbatim
86 *> WORK is REAL array, dimension (N)
87 *> \endverbatim
88 *>
89 *> \param[out] INFO
90 *> \verbatim
91 *> INFO is INTEGER
92 *> = 0: successful exit
93 *> < 0: if INFO = -i, the i-th argument had an illegal value
94 *> \endverbatim
95 *
96 * Authors:
97 * ========
98 *
99 *> \author Univ. of Tennessee
100 *> \author Univ. of California Berkeley
101 *> \author Univ. of Colorado Denver
102 *> \author NAG Ltd.
103 *
104 *> \date September 2012
105 *
106 *> \ingroup realPTcomputational
107 *
108 *> \par Further Details:
109 * =====================
110 *>
111 *> \verbatim
112 *>
113 *> The method used is described in Nicholas J. Higham, "Efficient
114 *> Algorithms for Computing the Condition Number of a Tridiagonal
115 *> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
116 *> \endverbatim
117 *>
118 * =====================================================================
119  SUBROUTINE sptcon( N, D, E, ANORM, RCOND, WORK, INFO )
120 *
121 * -- LAPACK computational routine (version 3.4.2) --
122 * -- LAPACK is a software package provided by Univ. of Tennessee, --
123 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
124 * September 2012
125 *
126 * .. Scalar Arguments ..
127  INTEGER info, n
128  REAL anorm, rcond
129 * ..
130 * .. Array Arguments ..
131  REAL d( * ), e( * ), work( * )
132 * ..
133 *
134 * =====================================================================
135 *
136 * .. Parameters ..
137  REAL one, zero
138  parameter( one = 1.0e+0, zero = 0.0e+0 )
139 * ..
140 * .. Local Scalars ..
141  INTEGER i, ix
142  REAL ainvnm
143 * ..
144 * .. External Functions ..
145  INTEGER isamax
146  EXTERNAL isamax
147 * ..
148 * .. External Subroutines ..
149  EXTERNAL xerbla
150 * ..
151 * .. Intrinsic Functions ..
152  INTRINSIC abs
153 * ..
154 * .. Executable Statements ..
155 *
156 * Test the input arguments.
157 *
158  info = 0
159  IF( n.LT.0 ) THEN
160  info = -1
161  ELSE IF( anorm.LT.zero ) THEN
162  info = -4
163  END IF
164  IF( info.NE.0 ) THEN
165  CALL xerbla( 'SPTCON', -info )
166  RETURN
167  END IF
168 *
169 * Quick return if possible
170 *
171  rcond = zero
172  IF( n.EQ.0 ) THEN
173  rcond = one
174  RETURN
175  ELSE IF( anorm.EQ.zero ) THEN
176  RETURN
177  END IF
178 *
179 * Check that D(1:N) is positive.
180 *
181  DO 10 i = 1, n
182  IF( d( i ).LE.zero )
183  $ RETURN
184  10 CONTINUE
185 *
186 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
187 *
188 * m(i,j) = abs(A(i,j)), i = j,
189 * m(i,j) = -abs(A(i,j)), i .ne. j,
190 *
191 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
192 *
193 * Solve M(L) * x = e.
194 *
195  work( 1 ) = one
196  DO 20 i = 2, n
197  work( i ) = one + work( i-1 )*abs( e( i-1 ) )
198  20 CONTINUE
199 *
200 * Solve D * M(L)**T * x = b.
201 *
202  work( n ) = work( n ) / d( n )
203  DO 30 i = n - 1, 1, -1
204  work( i ) = work( i ) / d( i ) + work( i+1 )*abs( e( i ) )
205  30 CONTINUE
206 *
207 * Compute AINVNM = max(x(i)), 1<=i<=n.
208 *
209  ix = isamax( n, work, 1 )
210  ainvnm = abs( work( ix ) )
211 *
212 * Compute the reciprocal condition number.
213 *
214  IF( ainvnm.NE.zero )
215  $ rcond = ( one / ainvnm ) / anorm
216 *
217  RETURN
218 *
219 * End of SPTCON
220 *
221  END