LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sptcon.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sptcon.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sptcon.f">
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*> \ingroup realPTcomputational
105*
106*> \par Further Details:
107* =====================
108*>
109*> \verbatim
110*>
111*> The method used is described in Nicholas J. Higham, "Efficient
112*> Algorithms for Computing the Condition Number of a Tridiagonal
113*> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
114*> \endverbatim
115*>
116* =====================================================================
117 SUBROUTINE sptcon( N, D, E, ANORM, RCOND, WORK, INFO )
118*
119* -- LAPACK computational routine --
120* -- LAPACK is a software package provided by Univ. of Tennessee, --
121* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
122*
123* .. Scalar Arguments ..
124 INTEGER INFO, N
125 REAL ANORM, RCOND
126* ..
127* .. Array Arguments ..
128 REAL D( * ), E( * ), WORK( * )
129* ..
130*
131* =====================================================================
132*
133* .. Parameters ..
134 REAL ONE, ZERO
135 parameter( one = 1.0e+0, zero = 0.0e+0 )
136* ..
137* .. Local Scalars ..
138 INTEGER I, IX
139 REAL AINVNM
140* ..
141* .. External Functions ..
142 INTEGER ISAMAX
143 EXTERNAL isamax
144* ..
145* .. External Subroutines ..
146 EXTERNAL xerbla
147* ..
148* .. Intrinsic Functions ..
149 INTRINSIC abs
150* ..
151* .. Executable Statements ..
152*
153* Test the input arguments.
154*
155 info = 0
156 IF( n.LT.0 ) THEN
157 info = -1
158 ELSE IF( anorm.LT.zero ) THEN
159 info = -4
160 END IF
161 IF( info.NE.0 ) THEN
162 CALL xerbla( 'SPTCON', -info )
163 RETURN
164 END IF
165*
166* Quick return if possible
167*
168 rcond = zero
169 IF( n.EQ.0 ) THEN
170 rcond = one
171 RETURN
172 ELSE IF( anorm.EQ.zero ) THEN
173 RETURN
174 END IF
175*
176* Check that D(1:N) is positive.
177*
178 DO 10 i = 1, n
179 IF( d( i ).LE.zero )
180 $ RETURN
181 10 CONTINUE
182*
183* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
184*
185* m(i,j) = abs(A(i,j)), i = j,
186* m(i,j) = -abs(A(i,j)), i .ne. j,
187*
188* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
189*
190* Solve M(L) * x = e.
191*
192 work( 1 ) = one
193 DO 20 i = 2, n
194 work( i ) = one + work( i-1 )*abs( e( i-1 ) )
195 20 CONTINUE
196*
197* Solve D * M(L)**T * x = b.
198*
199 work( n ) = work( n ) / d( n )
200 DO 30 i = n - 1, 1, -1
201 work( i ) = work( i ) / d( i ) + work( i+1 )*abs( e( i ) )
202 30 CONTINUE
203*
204* Compute AINVNM = max(x(i)), 1<=i<=n.
205*
206 ix = isamax( n, work, 1 )
207 ainvnm = abs( work( ix ) )
208*
209* Compute the reciprocal condition number.
210*
211 IF( ainvnm.NE.zero )
212 $ rcond = ( one / ainvnm ) / anorm
213*
214 RETURN
215*
216* End of SPTCON
217*
218 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sptcon(N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON
Definition: sptcon.f:118