Iteratively reweight least squares

Description

Uses the iteratively reweight least squares strategy to find an approximate L_p solution to Ax=b.

Usage

1
irls(A, b, tolr, tolx, p, maxiter)

Arguments

A

Matrix of the system of equations.

b

Right hand side of the system of equations

tolr

Tolerance below which residuals are ignored

tolx

Stopping tolerance. Stop when (norm(newx-x)/(1+norm(x)) < tolx)

p

Specifies which p-norm to use (most often, p=1.)

maxiter

Limit on number of iterations of IRLS

Details

Use to get L-1 norm solution of inverse problems.

Value

x

Approximate L_p solution

Author(s)

Jonathan M. Lees<jonathan.lees@unc.edu>

References

Aster, R.C., C.H. Thurber, and B. Borchers, Parameter Estimation and Inverse Problems, Elsevier Academic Press, Amsterdam, 2005.

Examples

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t = 1:10
y=c(109.3827,187.5385,267.5319,331.8753,386.0535,
428.4271,452.1644,498.1461,512.3499,512.9753)
sigma = rep(8, length(y))
N=length(t);

### % Introduce the outlier
y[4]=y[4]-200;

G = cbind( rep(1, N), t, -1/2*t^2 )

### % Apply the weighting

yw = y/sigma;

Gw = G/sigma

m2 = solve( t(Gw) %*% Gw , t(Gw) %*% yw, tol=1e-12 )


###  Solve for the 1-norm solution

m1 = irls(Gw,yw,1.0e-5,1.0e-5,1,25)
m1

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