LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cptcon.f
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1 *> \brief \b CPTCON
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPTCON( N, D, E, ANORM, RCOND, RWORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, N
25 * REAL ANORM, RCOND
26 * ..
27 * .. Array Arguments ..
28 * REAL D( * ), RWORK( * )
29 * COMPLEX E( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CPTCON computes the reciprocal of the condition number (in the
39 *> 1-norm) of a complex Hermitian positive definite tridiagonal matrix
40 *> using the factorization A = L*D*L**H or A = U**H*D*U computed by
41 *> CPTTRF.
42 *>
43 *> Norm(inv(A)) is computed by a direct method, and the reciprocal of
44 *> the condition number is computed as
45 *> RCOND = 1 / (ANORM * norm(inv(A))).
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] N
52 *> \verbatim
53 *> N is INTEGER
54 *> The order of the matrix A. N >= 0.
55 *> \endverbatim
56 *>
57 *> \param[in] D
58 *> \verbatim
59 *> D is REAL array, dimension (N)
60 *> The n diagonal elements of the diagonal matrix D from the
61 *> factorization of A, as computed by CPTTRF.
62 *> \endverbatim
63 *>
64 *> \param[in] E
65 *> \verbatim
66 *> E is COMPLEX array, dimension (N-1)
67 *> The (n-1) off-diagonal elements of the unit bidiagonal factor
68 *> U or L from the factorization of A, as computed by CPTTRF.
69 *> \endverbatim
70 *>
71 *> \param[in] ANORM
72 *> \verbatim
73 *> ANORM is REAL
74 *> The 1-norm of the original matrix A.
75 *> \endverbatim
76 *>
77 *> \param[out] RCOND
78 *> \verbatim
79 *> RCOND is REAL
80 *> The reciprocal of the condition number of the matrix A,
81 *> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
82 *> 1-norm of inv(A) computed in this routine.
83 *> \endverbatim
84 *>
85 *> \param[out] RWORK
86 *> \verbatim
87 *> RWORK is REAL array, dimension (N)
88 *> \endverbatim
89 *>
90 *> \param[out] INFO
91 *> \verbatim
92 *> INFO is INTEGER
93 *> = 0: successful exit
94 *> < 0: if INFO = -i, the i-th argument had an illegal value
95 *> \endverbatim
96 *
97 * Authors:
98 * ========
99 *
100 *> \author Univ. of Tennessee
101 *> \author Univ. of California Berkeley
102 *> \author Univ. of Colorado Denver
103 *> \author NAG Ltd.
104 *
105 *> \ingroup complexPTcomputational
106 *
107 *> \par Further Details:
108 * =====================
109 *>
110 *> \verbatim
111 *>
112 *> The method used is described in Nicholas J. Higham, "Efficient
113 *> Algorithms for Computing the Condition Number of a Tridiagonal
114 *> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
115 *> \endverbatim
116 *>
117 * =====================================================================
118  SUBROUTINE cptcon( N, D, E, ANORM, RCOND, RWORK, INFO )
119 *
120 * -- LAPACK computational routine --
121 * -- LAPACK is a software package provided by Univ. of Tennessee, --
122 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
123 *
124 * .. Scalar Arguments ..
125  INTEGER INFO, N
126  REAL ANORM, RCOND
127 * ..
128 * .. Array Arguments ..
129  REAL D( * ), RWORK( * )
130  COMPLEX E( * )
131 * ..
132 *
133 * =====================================================================
134 *
135 * .. Parameters ..
136  REAL ONE, ZERO
137  parameter( one = 1.0e+0, zero = 0.0e+0 )
138 * ..
139 * .. Local Scalars ..
140  INTEGER I, IX
141  REAL AINVNM
142 * ..
143 * .. External Functions ..
144  INTEGER ISAMAX
145  EXTERNAL isamax
146 * ..
147 * .. External Subroutines ..
148  EXTERNAL xerbla
149 * ..
150 * .. Intrinsic Functions ..
151  INTRINSIC abs
152 * ..
153 * .. Executable Statements ..
154 *
155 * Test the input arguments.
156 *
157  info = 0
158  IF( n.LT.0 ) THEN
159  info = -1
160  ELSE IF( anorm.LT.zero ) THEN
161  info = -4
162  END IF
163  IF( info.NE.0 ) THEN
164  CALL xerbla( 'CPTCON', -info )
165  RETURN
166  END IF
167 *
168 * Quick return if possible
169 *
170  rcond = zero
171  IF( n.EQ.0 ) THEN
172  rcond = one
173  RETURN
174  ELSE IF( anorm.EQ.zero ) THEN
175  RETURN
176  END IF
177 *
178 * Check that D(1:N) is positive.
179 *
180  DO 10 i = 1, n
181  IF( d( i ).LE.zero )
182  $ RETURN
183  10 CONTINUE
184 *
185 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
186 *
187 * m(i,j) = abs(A(i,j)), i = j,
188 * m(i,j) = -abs(A(i,j)), i .ne. j,
189 *
190 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
191 *
192 * Solve M(L) * x = e.
193 *
194  rwork( 1 ) = one
195  DO 20 i = 2, n
196  rwork( i ) = one + rwork( i-1 )*abs( e( i-1 ) )
197  20 CONTINUE
198 *
199 * Solve D * M(L)**H * x = b.
200 *
201  rwork( n ) = rwork( n ) / d( n )
202  DO 30 i = n - 1, 1, -1
203  rwork( i ) = rwork( i ) / d( i ) + rwork( i+1 )*abs( e( i ) )
204  30 CONTINUE
205 *
206 * Compute AINVNM = max(x(i)), 1<=i<=n.
207 *
208  ix = isamax( n, rwork, 1 )
209  ainvnm = abs( rwork( ix ) )
210 *
211 * Compute the reciprocal condition number.
212 *
213  IF( ainvnm.NE.zero )
214  $ rcond = ( one / ainvnm ) / anorm
215 *
216  RETURN
217 *
218 * End of CPTCON
219 *
220  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cptcon(N, D, E, ANORM, RCOND, RWORK, INFO)
CPTCON
Definition: cptcon.f:119