MotionGenesis: Solve nonlinear algebraic equations

Solve one nonlinear algebraic equation.

To numerically solve the equation x^2 - cos(x) = 0, type

Variable x Solve( x^2 - cos(x) = 0, x = 0.2 ) % x = 0.2 is a guess.

As shown in the plot to the right, this nonlinear equation has two solutions. In general, nonlinear equations have an unknown number of solutions (0, 1, 2, 9, ...). The Solve command frequently converges to a solution close to the starting guess.

If you guess	x = 0.2,	Solves to:	x = 0.82413
If you guess	x = -9,	Solves to:	x = -0.82413

To save commands (for subsequent re-use) or commands/responses, type

Save MGSolveNonlinearEqn.txt % Save commands. Save MGSolveNonlinearEqn.html % Save commands and responses.

Solve sets of nonlinear algebraic equations.

Equation for circle:	x² + y² = R²
Equation for sine curve:	y = A * sin(x)

To numerically solve the previous set of nonlinear equations for x and y, type

Constant R = 1 meter, A = 1 meter Variable x, y eqns[1] = x^2 + y^2 - R^2 % Circle eqns[2] = y - A*sin(x) % Sine wave Solve( eqns = 0, x = 3, y = 5 )

Note: The arguments x = 3 and y = 5 are a guess for the solution. The two solutions to these nonlinear equations depend on your guess, e.g.,

Guess	x = 3,	y = 5,	->	x = 0.7391,	y = 0.6736
Guess	x = -2,	y = -2,	->	x = -0.7391,	y = -0.6736

To save commands (for subsequent re-use) or commands/responses, type

Save MGSolveNonlinearEqns.txt % Commands. Save MGSolveNonlinearEqns.html % Commands and responses.

Continuous solution of one nonlinear algebraic equation.

y⁴ - 8*y - 0.3*t² - 9*sin(t) = 0 0 ≤ t ≤ 20

Program commands: MGSolveNonlinearEqnContinuously.txt
Program responses: MGSolveNonlinearEqnContinuously.html

Continuous solution of two coupled nonlinear algebraic equations for qA(t) and qB(t) for 0 ≤ t ≤ 2.2

cos(qA) + 2*cos(qB)	= 2 + 0.2*cos(t)
2*sin(qB) - sin(qA)	= 0.3sin(3t)

Program commands: MGSolveNonlinearEqnsContinuously.txt
Program responses: MGSolveNonlinearEqnsContinuously.html

Continuous solution of two coupled nonlinear algebraic equations for r(t) and θ(t) for 0 ≤ t ≤ 7.

x = r*cos(θ)	where x = cos(1.2*t)
y = r*sin(θ)	where y = sin(t)

Program commands: MGSolveNonlinearEqnsContinuouslyB.txt
Program responses: MGSolveNonlinearEqnsContinuouslyB.html

Advanced: Coding nonlinear algebraic equations.

Equation for circle:	x² + y² = R²
Equation for sine curve:	y = A * sin(x)

To write a MATLAB® code to numerically solve for x and y for 1 ≤ R ≤ 3, type

Constant R, A = 1 meter Variable x, y eqns[1] = x^2 + y^2 - R^2 % Circle eqns[2] = y - A*sin(x) % Sine wave Input x = 3 meters, y = 3 meters Output t seconds, x meters, y meters Nonlinear(eqns=0, x, y) [R = 1, 3, 0.5] CodeNonlinearEqns.m

MotionGenesis™ produces highly efficient and symbolically optimized codes.

C code CodeNonlinearEqns.c	Compile and link source code. Modify input values in CodeNonlinearEqns.in
Fortran code CodeNonlinearEqns.f	Compile and link source code. Modify input values in CodeNonlinearEqns.in
MATLAB® CodeNonlinearEqns.m	Invoke MATLAB® and type CodeNonlinearEqns Modify input values in CodeNonlinearEqns.m

Running the MotionGenesis™-generated .m file produces, e.g.,

% FILE: CodeNonlinearEqns.1
%
% Residual     R         x          y
% (UNITS)   (meters)  (meters)   (meters)

1.221E-15  1.00E+00  7.391E-01  6.736E-01
1.177E-14  1.50E+00  1.181E+00  9.249E-01
3.483E-13  2.00E+00  1.740E+00  9.857E-01
2.209E-13  2.50E+00  2.409E+00  6.689E-01
4.656E-15  3.00E+00  2.997E+00  1.446E-01