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/* eigens.c
*
* Eigenvalues and eigenvectors of a real symmetric matrix
*
*
*
* SYNOPSIS:
*
* int n;
* double A[n*(n+1)/2], EV[n*n], E[n];
* void eigens( A, EV, E, n );
*
*
*
* DESCRIPTION:
*
* The algorithm is due to J. vonNeumann.
* - -
* A[] is a symmetric matrix stored in lower triangular form.
* That is, A[ row, column ] = A[ (row*row+row)/2 + column ]
* or equivalently with row and column interchanged. The
* indices row and column run from 0 through n-1.
*
* EV[] is the output matrix of eigenvectors stored columnwise.
* That is, the elements of each eigenvector appear in sequential
* memory order. The jth element of the ith eigenvector is
* EV[ n*i+j ] = EV[i][j].
*
* E[] is the output matrix of eigenvalues. The ith element
* of E corresponds to the ith eigenvector (the ith row of EV).
*
* On output, the matrix A will have been diagonalized and its
* orginal contents are destroyed.
*
* ACCURACY:
*
* The error is controlled by an internal parameter called RANGE
* which is set to 1e-10. After diagonalization, the
* off-diagonal elements of A will have been reduced by
* this factor.
*
* ERROR MESSAGES:
*
* None.
*
*/
/*
Copyright 1973, 1991 by Stephen L. Moshier
Copyleft version.
*/
void eigens( A, RR, E, N )
double A[], RR[], E[];
int N;
{
int IND, L, LL, LM, M, MM, MQ, I, J, K, IA, LQ;
int IQ, IM, IL, NLI, NMI;
double ANORM, ANORMX, AIA, THR, ALM, QI, ALL, AMM, X, Y;
double SINX, SINX2, COSX, COSX2, SINCS, AIL, AIM;
double RLI, RMI, Q, V;
double sqrt(), fabs();
static double RANGE = 1.0e-10; /*3.0517578e-5;*/
/* Initialize identity matrix in RR[] */
for( J=0; J<N*N; J++ )
RR[J] = 0.0;
MM = 0;
for( J=0; J<N; J++ )
{
RR[MM + J] = 1.0;
MM += N;
}
ANORM=0.0;
for( I=0; I<N; I++ )
{
for( J=0; J<N; J++ )
{
if( I != J )
{
IA = I + (J*J+J)/2;
AIA = A[IA];
ANORM += AIA * AIA;
}
}
}
if( ANORM <= 0.0 )
goto done;
ANORM = sqrt( ANORM + ANORM );
ANORMX = ANORM * RANGE / N;
THR = ANORM;
while( THR > ANORMX )
{
THR=THR/N;
do
{ /* while IND != 0 */
IND = 0;
for( L=0; L<N-1; L++ )
{
for( M=L+1; M<N; M++ )
{
MQ=(M*M+M)/2;
LM=L+MQ;
ALM=A[LM];
if( fabs(ALM) < THR )
continue;
IND=1;
LQ=(L*L+L)/2;
LL=L+LQ;
MM=M+MQ;
ALL=A[LL];
AMM=A[MM];
X=(ALL-AMM)/2.0;
Y=-ALM/sqrt(ALM*ALM+X*X);
if(X < 0.0)
Y=-Y;
SINX = Y / sqrt( 2.0 * (1.0 + sqrt( 1.0-Y*Y)) );
SINX2=SINX*SINX;
COSX=sqrt(1.0-SINX2);
COSX2=COSX*COSX;
SINCS=SINX*COSX;
/* ROTATE L AND M COLUMNS */
for( I=0; I<N; I++ )
{
IQ=(I*I+I)/2;
if( (I != M) && (I != L) )
{
if(I > M)
IM=M+IQ;
else
IM=I+MQ;
if(I >= L)
IL=L+IQ;
else
IL=I+LQ;
AIL=A[IL];
AIM=A[IM];
X=AIL*COSX-AIM*SINX;
A[IM]=AIL*SINX+AIM*COSX;
A[IL]=X;
}
NLI = N*L + I;
NMI = N*M + I;
RLI = RR[ NLI ];
RMI = RR[ NMI ];
RR[NLI]=RLI*COSX-RMI*SINX;
RR[NMI]=RLI*SINX+RMI*COSX;
}
X=2.0*ALM*SINCS;
A[LL]=ALL*COSX2+AMM*SINX2-X;
A[MM]=ALL*SINX2+AMM*COSX2+X;
A[LM]=(ALL-AMM)*SINCS+ALM*(COSX2-SINX2);
} /* for M=L+1 to N-1 */
} /* for L=0 to N-2 */
}
while( IND != 0 );
} /* while THR > ANORMX */
done: ;
/* Extract eigenvalues from the reduced matrix */
L=0;
for( J=1; J<=N; J++ )
{
L=L+J;
E[J-1]=A[L-1];
}
}
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