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# Lanczos algorithm to compute eigen vectors and eigen values of a square matrix

A = read($1); # input data, must be a square matrix

m = nrow(A);
v0 = matrix(0, m, 1);
v1 = Rand(rows=m, cols=1);
v1 = sqrt(v1/sum(v1)); # vector with norm 1

T = matrix(0, m, m);
TV = matrix(0, m, m);

# Lanczos iterations to compute Tri-Diagonal Matrix
beta = 0;
for(i in seq(1,m)) {
    if (i == m) {
        w1 = A %*% v1;
        alpha = sum(w1*v1);
        T[m, m] = alpha;
    } else {
        w1 = A %*% v1;
        alpha = sum(w1*v1);
        w1 = w1 - alpha*v1 - beta*v0;
        beta = sqrt(sum(w1^2));
        v0 = v1;
        v1 = w1/beta

        T[i, i] = alpha;
        T[i+1, i] = beta;
        T[i, i+1] = beta;
    }
    TV[,i] = v1;
}
# compute eigen vectors of Tri-diagonal matrix
[eval, tevec] = eigen(T)
# compute eigen vectors of Tri-diagonal matrix
evec = TV %*% tevec;

# output data
write(eval, $2);
write(evec, $3);