Find the least squares for a machine-precision matrix:

Least squares for a complex matrix:

Use LeastSquares for an exact non-square matrix:

Least squares for an arbitrary-precision matrix:

Use LeastSquares with a symbolic matrix:

The least squares for a large numerical matrix is computed efficiently:

In LeastSquares[m,b], b can be a matrix:

Each column in the result equals the solution found by using the corresponding column in b as input:

Solve a least-squares problem for a sparse matrix:

Solve the least-squares problem with structured matrices:

Use a different type of matrix structure:

LeastSquares[IdentityMatrix[n],b] gives the vector :

Least squares of HilbertMatrix:

Solve a least-squares problem with a 2×3×4 array and a 2×3×5×6 array :

The result is a 4×5×6 array:

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:

LeastSquares[m,b] can be understood as finding the solution to , where is the orthogonal projection of onto the column space of . Consider the following and :

Find an orthonormal basis for the space spanned by the columns of :

Compute the orthogonal projection of onto the spaced spanned by the :

Visualize , its projections onto the , and :

Solve :

This is the same result as given by LeastSquares:

Compare and explain the answers returned by LeastSquares[m,b] and LinearSolve[m,b^⟂] for the following and :

Find an orthonormal basis for the space spanned by the columns of :

A zero vector is returned because the rank of the matrix is less than the number of columns:

Compute the orthogonal projection of onto the spaced spanned by the :

Solve :

Find the solution returned by LeastSquares:

While x and xPerp are different, both solve the least-squares problem because m.x==m.xPerp:

The two solutions differ by an element of NullSpace[m]:

Use the matrix projection operators for a matrix with linearly independent columns to find LeastSquares[m,b] for the following and :

The projection operator on the column space of is :

The solution to the least-squares problem is then the unique solution to :

Confirm using LeastSquares:

Compare the solutions found using LeastSquares[m,b] and LinearSolve together with the normal equations of and for the following and :

Solve using LeastSquares:

Solve using LinearSolve and the normal equations :

While x and xNormal are different, both solve the least-squares problem because m.x==m.xNormal:

The two solutions differ by an element of NullSpace[m]:

LeastSquares can be used to find a best-fit curve to data. Consider the following data:

Extract the and coordinates from the data:

Let have the columns and , so that minimizing will be fitting to a line :

Get the coefficients and for a linear least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find the best-fit parabola to the following data:

Extract the and coordinates from the data:

Let have the columns , and , so that minimizing will be fitting to :

Get the coefficients , and for a least‐squares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

A healthy child’s systolic blood pressure (in millimeters of mercury) and weight (in pounds) are approximately related by the equation . Use the following experimental data points to estimate the systolic blood pressure of a healthy child weighing 100 pounds:

Use DesignMatrix to construct the matrix with columns and :

Extract the values from the data:

The least-squares solution of :

Substitute the parameters into the model:

Then the expected blood pressure of a child weighing 100 pounds is roughly :

Visualize the best-fit curve and the data:

According to Kepler’s first law, a comet's orbit satisfies , where is a constant and is the eccentricity. The eccentricity determines the type of orbit, with for an ellipse, for a parabola, and for a hyperbola. Use the following observational data to determine the type of orbit of the comet and predict its distance from the Sun at :

To find and , first use DesignMatrix to create the matrix whose columns are and :

Use LeastSquares to find the and that minimize the error in from the design matrix:

Since , the orbit is elliptical and there is a unique value of for each value of :

Evaluating the function at gives an expected distance of roughly :

Consider the following data:

Extract the and coordinates from the 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:

Visualize the data with the best-fit piecewise cubic, which is :