A library that provides routines to compute the solutions to systems of nonlinear equations.
Documentation can be found here
CMakeThis library can be built using CMake. For instructions see Running CMake.
FPM can also be used to build this library using the provided fpm.toml.
fpm buildThe NONLIN library can be used within your FPM project by adding the following to your fpm.toml file.
[dependencies]
nonlin = { git = "https://github.com/jchristopherson/nonlin" }Here is a list of external code libraries utilized by this library.
This example solves a set of two equations of two unknowns using a Quasi-Newton type solver. In this example, the solver is left to compute the derivatives numerically.
program example
use iso_fortran_env
use nonlin_core
use nonlin_solve, only : quasi_newton_solver
implicit none
! Local Variables
type(vecfcn_helper) :: obj
procedure(vecfcn), pointer :: fcn
type(iteration_behavior) :: ib
type(quasi_newton_solver) :: solver
real(real64) :: x(2), f(2)
! Locate the routine containing the equations to solve
fcn => fcns
call obj%set_fcn(fcn, 2, 2)
! Define an initial guess
x = 1.0d0 ! Equivalent to x = [1.0d0, 1.0d0]
! Defining solver parameters. This step is optional as the defaults are
! typically sufficient; however, this is being done for illustration
! purposes.
!
! Establish how many iterations are allowed to pass before the solver
! forces a re-evaluation of the Jacobian matrix. Notice, the solver may
! choose to re-evaluate the Jacobian sooner than this, but that is
! dependent upon the behavior of the problem.
call solver%set_jacobian_interval(20)
! Establish convergence criteria. Again, this step is optional as the
! defaults are typically sufficient; however, this is being done for
! illustration purposes.
call solver%set_fcn_tolerance(1.0d-8)
call solver%set_var_tolerance(1.0d-12)
call solver%set_gradient_tolerance(1.0d-12)
! Solve
call solver%solve(obj, x, f, ib)
! Display the output
print '(AF7.5AF7.5A)', "Solution: (", x(1), ", ", x(2), ")"
print '(AE9.3AE9.3A)', "Residual: (", f(1), ", ", f(2), ")"
print '(AI0)', "Iterations: ", ib%iter_count
print '(AI0)', "Function Evaluations: ", ib%fcn_count
print '(AI0)', "Jacobian Evaluations: ", ib%jacobian_count
contains
! Define the routine containing the equations to solve. The equations are:
! x**2 + y**2 = 34
! x**2 - 2 * y**2 = 7
subroutine fcns(x, f)
real(real64), intent(in), dimension(:) :: x
real(real64), intent(out), dimension(:) :: f
f(1) = x(1)**2 + x(2)**2 - 34.0d0
f(2) = x(1)**2 - 2.0d0 * x(2)**2 - 7.0d0
end subroutine
end programThe above program produces the following output.
Solution: (5.00000, 3.00000)
Residual: (0.323E-11, 0.705E-11)
Iterations: 11
Function Evaluations: 15
Jacobian Evaluations: 1
This example uses a least-squares approach to determine the coefficients of a polynomial that best fits a set of data.
program example
use iso_fortran_env
use nonlin_core
use nonlin_least_squares, only : least_squares_solver
implicit none
! Local Variables
type(vecfcn_helper) :: obj
procedure(vecfcn), pointer :: fcn
type(least_squares_solver) :: solver
real(real64) :: x(4), f(21) ! There are 4 coefficients and 21 data points
! Locate the routine containing the equations to solve
fcn => fcns
call obj%set_fcn(fcn, 21, 4)
! Define an initial guess
x = 1.0d0 ! Equivalent to x = [1.0d0, 1.0d0, 1.0d0, 1.0d0]
! Solve
call solver%solve(obj, x, f)
! Display the output
print '(AF12.10)', "c0: ", x(4)
print '(AF12.10)', "c1: ", x(3)
print '(AF12.10)', "c2: ", x(2)
print '(AF12.10)', "c3: ", x(1)
print '(AF7.5)', "Max Residual: ", maxval(abs(f))
contains
! The function containing the data to fit
subroutine fcns(x, f)
! Arguments
real(real64), intent(in), dimension(:) :: x ! Contains the coefficients
real(real64), intent(out), dimension(:) :: f
! Local Variables
real(real64), dimension(21) :: xp, yp
! Data to fit (21 data points)
xp = [0.0d0, 0.1d0, 0.2d0, 0.3d0, 0.4d0, 0.5d0, 0.6d0, 0.7d0, 0.8d0, &
0.9d0, 1.0d0, 1.1d0, 1.2d0, 1.3d0, 1.4d0, 1.5d0, 1.6d0, 1.7d0, &
1.8d0, 1.9d0, 2.0d0]
yp = [1.216737514d0, 1.250032542d0, 1.305579195d0, 1.040182335d0, &
1.751867738d0, 1.109716707d0, 2.018141531d0, 1.992418729d0, &
1.807916923d0, 2.078806005d0, 2.698801324d0, 2.644662712d0, &
3.412756702d0, 4.406137221d0, 4.567156645d0, 4.999550779d0, &
5.652854194d0, 6.784320119d0, 8.307936836d0, 8.395126494d0, &
10.30252404d0]
! We'll apply a cubic polynomial model to this data:
! y = c3 * x**3 + c2 * x**2 + c1 * x + c0
f = x(1) * xp**3 + x(2) * xp**2 + x(3) * xp + x(4) - yp
! For reference, the data was generated by adding random errors to
! the following polynomial: y = x**3 - 0.3 * x**2 + 1.2 * x + 0.3
end subroutine
end programThe above program produces the following output.
c0: 1.1866142244
c1: 0.4466134462
c2: -.1223202909
c3: 1.0647627571
Max Residual: 0.50636
The following graph illustrates the fit.
This example utilizes the polynomial type to fit a polynomial to the data set utilized in Example 2.
program example
use iso_fortran_env
use nonlin_polynomials
implicit none
! Local Variables
integer(int32) :: i
real(real64), dimension(21) :: xp, yp, yf, yc, err
real(real64) :: res
type(polynomial) :: p
! Data to fit
xp = [0.0d0, 0.1d0, 0.2d0, 0.3d0, 0.4d0, 0.5d0, 0.6d0, 0.7d0, 0.8d0, &
0.9d0, 1.0d0, 1.1d0, 1.2d0, 1.3d0, 1.4d0, 1.5d0, 1.6d0, 1.7d0, &
1.8d0, 1.9d0, 2.0d0]
yp = [1.216737514d0, 1.250032542d0, 1.305579195d0, 1.040182335d0, &
1.751867738d0, 1.109716707d0, 2.018141531d0, 1.992418729d0, &
1.807916923d0, 2.078806005d0, 2.698801324d0, 2.644662712d0, &
3.412756702d0, 4.406137221d0, 4.567156645d0, 4.999550779d0, &
5.652854194d0, 6.784320119d0, 8.307936836d0, 8.395126494d0, &
10.30252404d0]
! Create a copy of yp as it will be overwritten in the fit command
yc = yp
! Fit the polynomial
call p%fit(xp, yp, 3)
! Evaluate the polynomial at xp, and then determine the residual
yf = p%evaluate(xp)
err = abs(yf - yc)
res = maxval(err)
! Print out the coefficients
print '(AI0AF12.10)', ("c", i - 1, " = ", p%get(i), i = 1, 4)
print '(AF7.5)', "Max Residual: ", res
end programThe above program yields the following coefficients.
c0 = 1.1866141861
c1 = 0.4466136311
c2 = -.1223204989
c3 = 1.0647628218
Max Residual: 0.50636
Notice, as expected, the results are very similar to the output of Example 2.
This example uses the Nelder-Mead simplex method to find the minimum of the Rosenbrock function.
program example
use iso_fortran_env
use nonlin_optimize, only : nelder_mead
use nonlin_core
implicit none
! Local Variables
type(nelder_mead) :: solver
type(fcnnvar_helper) :: obj
procedure(fcnnvar), pointer :: fcn
real(real64) :: x(2), fout
type(iteration_behavior) :: ib
! Initialization
fcn => rosenbrock
call obj%set_fcn(fcn, 2)
! Define an initial guess - the solution is (1, 1)
call random_number(x)
! Call the solver
call solver%solve(obj, x, fout, ib)
! Display the output
print '(AF7.5AF7.5A)', "Minimum: (", x(1), ", ", x(2), ")"
print '(AE9.3)', "Function Value: ", fout
print '(AI0)', "Iterations: ", ib%iter_count
print '(AI0)', "Function Evaluations: ", ib%fcn_count
contains
! Rosenbrock's Function
function rosenbrock(x) result(f)
real(real64), intent(in), dimension(:) :: x
real(real64) :: f
f = 1.0d2 * (x(2) - x(1)**2)**2 + (x(1) - 1.0d0)**2
end function
end programThe above program produces the following output:
Minimum: (1.00000, 1.00000)
Function Value: 0.121E-12
Iterations: 52
Function Evaluations: 101
Notice, the convergence tolerance was set to its default value (1e-12).