LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dptcon()

 subroutine dptcon ( integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer INFO )

DPTCON

Purpose:
``` DPTCON computes the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite tridiagonal matrix
using the factorization A = L*D*L**T or A = U**T*D*U computed by
DPTTRF.

Norm(inv(A)) is computed by a direct method, and the reciprocal of
the condition number is computed as
RCOND = 1 / (ANORM * norm(inv(A))).```
Parameters
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization of A, as computed by DPTTRF.``` [in] E ``` E is DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization of A, as computed by DPTTRF.``` [in] ANORM ``` ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.``` [out] RCOND ``` RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the 1-norm of inv(A) computed in this routine.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The method used is described in Nicholas J. Higham, "Efficient
Algorithms for Computing the Condition Number of a Tridiagonal
Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.```

Definition at line 117 of file dptcon.f.

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 DOUBLE PRECISION ANORM, RCOND
126* ..
127* .. Array Arguments ..
128 DOUBLE PRECISION D( * ), E( * ), WORK( * )
129* ..
130*
131* =====================================================================
132*
133* .. Parameters ..
134 DOUBLE PRECISION ONE, ZERO
135 parameter( one = 1.0d+0, zero = 0.0d+0 )
136* ..
137* .. Local Scalars ..
138 INTEGER I, IX
139 DOUBLE PRECISION AINVNM
140* ..
141* .. External Functions ..
142 INTEGER IDAMAX
143 EXTERNAL idamax
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( 'DPTCON', -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 = idamax( 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 DPTCON
217*
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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