# Jacobian Multiply Function with Linear Least Squares

Using a Jacobian multiply function, you can solve a least-squares problem of the form

`$\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}$`

such that `lb ≤ x ≤ ub`, for problems where C is very large, perhaps too large to be stored. For this technique, use the `'trust-region-reflective'` algorithm.

For example, consider a problem where C is a 2n-by-n matrix based on a circulant matrix. The rows of C are shifts of a row vector v. This example has the row vector v with elements of the form $\left(–1{\right)}^{k+1}/k$:

$v=\left[1,-1/2,1/3,-1/4,\dots ,-1/n\right]$,

where the elements are cyclically shifted.

`$C=\left[\begin{array}{ccccc}1& -1/2& 1/3& ...& -1/n\\ -1/n& 1& -1/2& ...& 1/\left(n-1\right)\\ 1/\left(n-1\right)& -1/n& 1& ...& -1/\left(n-2\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ -1/2& 1/3& -1/4& ...& 1\\ 1& -1/2& 1/3& ...& -1/n\\ -1/n& 1& -1/2& ...& 1/\left(n-1\right)\\ 1/\left(n-1\right)& -1/n& 1& ...& -1/\left(n-2\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ -1/2& 1/3& -1/4& ...& 1\end{array}\right].$`

This least-squares example considers the problem where

$d=\left[n-1,n-2,\dots ,-n\right]$,

and the constraints are $-5\le {x}_{i}\le 5$ for $i=1,\dots ,n$.

For large enough $n$, the dense matrix C does not fit into computer memory ($n=10,000$ is too large on one tested system).

A Jacobian multiply function has the following syntax.

`w = jmfcn(Jinfo,Y,flag)`

`Jinfo` is a matrix the same size as C, used as a preconditioner. If C is too large to fit into memory, `Jinfo` should be sparse. `Y` is a vector or matrix sized so that `C*Y` or `C'*Y` works as matrix multiplication. `flag` tells `jmfcn` which product to form:

• `flag` > 0 ⇒  `w = C*Y`

• `flag` < 0 ⇒  `w = C'*Y`

• `flag` = 0 ⇒  `w = C'*C*Y`

Because C is such a simply structured matrix, you can easily write a Jacobian multiply function in terms of the vector v; that is, without forming C. Each row of `C*Y` is the product of a circularly shifted version of v times `Y`. Use `circshift` to circularly shift v.

To compute `C*Y`, compute `v*Y` to find the first row, then shift v and compute the second row, and so on.

To compute `C'*Y`, perform the same computation, but use a shifted version of `temp`, the vector formed from the first row of `C'`:

`temp = [fliplr(v),fliplr(v)];`

`temp = [circshift(temp,1,2),circshift(temp,1,2)]; % Now temp = C'(1,:)`

To compute `C'*C*Y`, simply compute `C*Y` using shifts of v, and then compute `C'` times the result using shifts of `fliplr(v)`.

The helper function `lsqcirculant3` is a Jacobian multiply function that implements this procedure; it appears at the end of this example.

The `dolsqJac3` helper function at the end of this example sets up the vector v and calls the solver `lsqlin` using the `lsqcirculant3` Jacobian multiply function.

When `n` = 3000, C is an 18,000,000-element dense matrix. Determine the results of the `dolsqJac3` function for `n` = 3000 at selected values of x, and display the output structure.

`[x,resnorm,residual,exitflag,output] = dolsqJac3(3000);`
```Local minimum possible. lsqlin stopped because the relative change in function value is less than the function tolerance. ```
`disp(x(1))`
``` 5.0000 ```
`disp(x(1500))`
``` -0.5201 ```
`disp(x(3000))`
``` -5.0000 ```
`disp(output)`
``` iterations: 16 algorithm: 'trust-region-reflective' firstorderopt: 5.9351e-05 cgiterations: 36 constrviolation: [] linearsolver: [] message: 'Local minimum possible.↵↵lsqlin stopped because the relative change in function value is less than the function tolerance.' ```

### Helper Functions

This code creates the `lsqcirculant3` helper function.

```function w = lsqcirculant3(Jinfo,Y,flag,v) % This function computes the Jacobian multiply function % for a 2n-by-n circulant matrix example if flag > 0 w = Jpositive(Y); elseif flag < 0 w = Jnegative(Y); else w = Jnegative(Jpositive(Y)); end function a = Jpositive(q) % Calculate C*q temp = v; a = zeros(size(q)); % Allocating the matrix a a = [a;a]; % The result is twice as tall as the input. for r = 1:size(a,1) a(r,:) = temp*q; % Compute the rth row temp = circshift(temp,1,2); % Shift the circulant end end function a = Jnegative(q) % Calculate C'*q temp = fliplr(v); temp = circshift(temp,1,2); % Shift the circulant for C' len = size(q,1)/2; % The returned vector is half as long % as the input vector. a = zeros(len,size(q,2)); % allocating the matrix a for r = 1:len a(r,:) = [temp,temp]*q; % Compute the rth row temp = circshift(temp,1,2); % Shift the circulant end end end```

This code creates the `dolsqJac3` helper function.

```function [x,resnorm,residual,exitflag,output] = dolsqJac3(n) % r = 1:n-1; % Index for making vectors v(n) = (-1)^(n+1)/n; % Allocating the vector v v(r) =( -1).^(r+1)./r; % Now C should be a 2n-by-n circulant matrix based on v, % but it might be too large to fit into memory. r = 1:2*n; d(r) = n-r; Jinfo = [speye(n);speye(n)]; % Sparse matrix for preconditioning % This matrix is a required input for the solver; % preconditioning is not used in this example. % Pass the vector v so that it does not need to be % computed in the Jacobian multiply function. options = optimoptions('lsqlin','Algorithm','trust-region-reflective',... 'JacobianMultiplyFcn',@(Jinfo,Y,flag)lsqcirculant3(Jinfo,Y,flag,v)); lb = -5*ones(1,n); ub = 5*ones(1,n); [x,resnorm,residual,exitflag,output] = ... lsqlin(Jinfo,d,[],[],[],[],lb,ub,[],options); end```