# tikhonov.m

Tikhonov regularised solution to K*x=y with a positivity constraint on x using regularised Newton-Raphson method.

## Syntax

[x,err,reg]=tikhonov(K,D,KtK,DtD,H,y,lambda)

## Arguments

K - kernel matrix, may be complex, may be non-square D - regularisation matrix, leave empty to use finite difference second derivative matrix KtK - K'*K, for repeated calls it may be faster to pre- compute this quantity, leave empty otherwise DtD - D'*D, for repeated calls it may be faster to pre- compute this quantity, leave empty otherwise H - Tikhonov Hessian 2*real(KtK+lambda*DtD), for re- peated calls it may be faster to precompute this quantity, leave empty otherwise y - a column vector, may be complex lambda - Tikhonov regularisation parameter

## Outputs

x - a real vector, a minimum (subject to positivity) of norm(K*x-y,2)^2+lambda*norm(D*x,2)^2 err - error signal norm(K*x-y,2)^2 reg - regularisation signal norm(D*x,2)^2

## Notes

For best numerical performance, scale K to have approximately unit 2-norm, and y to have approximately unit 1-norm.

## See also

*Version 2.8, authors: Anupama Acharya, Ilya Kuprov*