Built-in Mathematica Symbol

LeastSquares

LeastSquares[m, b]
finds an x which solves the linear least-squares problem for the matrix equation m.x

b.

MORE INFORMATION

LeastSquares[m, b] gives a vector x which minimizes Norm[m.x-b].

The vector x is uniquely determined by the minimization only if Length[x]MatrixRank[m].

The argument b can be a matrix, in which case the least-squares minimization is done independently for each column in b.

LeastSquares works on both numerical and symbolic matrices, as well as SparseArray objects.

A Method option can also be given. Settings for arbitrary-precision numerical matrices include "Direct" and "IterativeRefinement", and for sparse arrays "Direct" and "Krylov". The default setting of Automatic switches between these methods depending on the matrix given.

EXAMPLES

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Basic Examples (1)

Solve a simple least-squares problem:

In[1]:=

Out[1]=

Scope (4)

Use symbolic input:

m is a 3×4 matrix and b is a length-4 vector:

Use exact arithmetic to find a vector x that minimizes

:

Use machine arithmetic:

Use 20-digit precision arithmetic:

Solve the least-squares problem for a random complex matrix:

Use a sparse matrix:

Generalizations & Extensions (1)

b can be a matrix:

The first column of the b matrix is used to generate the first column of the result:

Options (1)

m is a 20×20 Hilbert matrix and b is a vector such that the solution of m.x = b is known:

With the default tolerance, numerical roundoff is limited so errors are distributed:

With Tolerance->0, numerical roundoff can introduce excessive error:

Specifying a higher tolerance will limit roundoff errors at the expense of a larger residual:

Applications (1)

Here is some data:

Define cubic basis functions centered at t with support on the interval [t - 2, t + 2]:

Set up a sparse design matrix for basis functions centered at 0, 1, ..., 10:

Solve the least-squares problem:

Properties & Relations (3)

When m.x

b can be solved exactly, LeastSquares is equivalent to LinearSolve:

m is a 5×2 matrix and b is a length-5 vector:

Solve the least-squares problem:

This is the minimizer of

:

It is also gives the coefficients for the line with least-squares distance to the points:

LeastSquares gives the parameter estimates for a linear model with normal errors:

LinearModelFit fits the model and gives additional information about the fitting:

The parameter estimates:

Extract additional results: