75.1 Introduction to minpack

Minpack is a Common Lisp translation (via f2cl) of the Fortran library MINPACK as obtained from Netlib.

75.2 Functions and Variables for minpack

Function: minpack_lsquares (flist, varlist, guess, ['tolerance = tolerance, 'jacobian = jacobian]) ¶

Compute the point that minimizes the sum of the squares of the functions in the list flist. The variables are in the list varlist. An initial guess of the optimum point must be provided in guess.

Let flist be a list of m functions, \(f_i(x_1, x_2, ..., x_n).\) Then this function can be used to find the values of \(x_1, x_2, ..., x_n\) that solve the least squares problem

$$ \sum_i^m f_i(x_1, x_2,...,x_n)^2 $$

The optional keyword arguments, tolerance and jacobian provide some control over the algorithm.

tolerance

the estimated relative error desired in the sum of squares. The default value is approximately \(1.0537\times 10^{-8}.\)

jacobian

specifies the Jacobian. If jacobian is not given or is true (the default), the Jacobian is computed from flist. If jacobian is false, a numerical approximation is used. See Jacobian.

minpack_lsquares returns a list of three items as follows:

1. The estimated solution
2. The sum of squares
3. The success of the algorithm. The possible values are
   0. improper input parameters.
   1. algorithm estimates that the relative error in the sum of squares is at most tolerance.
   2. algorithm estimates that the relative error between x and the solution is at most tolerance.
   3. conditions for info = 1 and info = 2 both hold.
   4. fvec is orthogonal to the columns of the jacobian to machine precision.
   5. number of calls to fcn with iflag = 1 has reached 100*(n+1).
   6. tol is too small. no further reduction in the sum of squares is possible.
   7. tol is too small. no further improvement in the approximate solution x is possible.

Here is an example using Powell’s singular function.

(%i1) load("minpack")$
(%i2) powell(x1,x2,x3,x4) := [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2, sqrt(10)*(x1-x4)^2]$

(%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4], [3,-1,0,1]);
(%o3) [[1.6521175961683935e-17, - 1.6521175961683934e-18, 
2.6433881538694683e-18, 2.6433881538694683e-18], 
6.109327859207777e-34, 4]

Same problem but use numerical approximation to Jacobian.

(%i1) load("minpack")$
(%i2) powell(x1,x2,x3,x4) := [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2, sqrt(10)*(x1-x4)^2]$

(%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4], [3,-1,0,1], jacobian = false);
(%o3) [[5.060282149485331e-11, - 5.060282149491206e-12, 
2.1794478435472183e-11, 2.1794478435472183e-11], 
3.534491794847031e-21, 5]

Categories: Package minpack ·

Function: minpack_solve (flist, varlist, guess, ['tolerance = tolerance, 'jacobian = jacobian]) ¶

Solve a system of n equations in n unknowns. The n equations are given in the list flist, and the unknowns are in varlist. An initial guess of the solution must be provided in guess.

Let flist be a list of m functions, \(f_i(x_1, x_2, ..., x_n).\) Then this functions solves the system of m nonlinear equations in n variables:

$$ f_i(x_1, x_2, ..., x_n) = 0 $$

The optional keyword arguments, tolerance and jacobian provide some control over the algorithm.

tolerance

the estimated relative error desired in the sum of squares. The default value is approximately \(1.0537\times 10^{-8}.\)

jacobian

specifies the Jacobian. If jacobian is not given or is true (the default), the Jacobian is computed from flist. If jacobian is false, a numerical approximation is used. See Jacobian.

minpack_solve returns a list of three items as follows:

1. The estimated solution
2. The sum of squares
3. The success of the algorithm. The possible values are
   0. improper input parameters.
   1. algorithm estimates that the relative error in the solution is at most tolerance.
   2. number of calls to fcn with iflag = 1 has reached 100*(n+1).
   3. tol is too small. no further reduction in the sum of squares is possible.
   4. Iteration is not making good progress.

(%i1) load("minpack")$
(%i2) powell(x1,x2,x3,x4) := [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2, sqrt(10)*(x1-x4)^2]$

(%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4], [3,-1,0,1]);
(%o3) [[1.6521175961683935e-17, - 1.6521175961683934e-18, 
2.6433881538694683e-18, 2.6433881538694683e-18], 
6.109327859207777e-34, 4]

In this particular case, we can solve this analytically:

(%i1) powell(x1,x2,x3,x4) := [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2, sqrt(10)*(x1-x4)^2]$

(%i2) solve(powell(x1,x2,x3,x4),[x1,x2,x3,x4]);
(%o2)          [[x1 = 0, x2 = 0, x3 = 0, x4 = 0]]

and we see that the numerical solution is quite close the analytical one.

Categories: Package minpack ·