Matlab Symbolic Toolbox Solve System Of Equations

[Lucrecio Poinson]

Tocircumvent this difficulty, solve can return a structure whose fields are the solutions. For example, solve the system of equations u^2 - v^2 = a^2, u + v = 1, a^2 - 2*a = 3. The solver returns its results enclosed in a structure.

Similar comments apply to the solutions for u and v. The structure S can now be manipulated by the field and index to access a particular portion of the solution. For example, to examine the second solution, you can use the following statement to extract the second component of each field.

solve does not automatically return all solutions of an equation. To return all solutions along with the parameters in the solution and the conditions on the solution, set the ReturnConditions option to true.

Visualize the system of equations using fimplicit. To set the x-axis and y-axis values in terms of pi, get the axes handles using axes in a. Create the symbolic array S of the values -2*pi to 2*pi at intervals of pi/2. To set the ticks to S, use the XTick and YTick properties of a. To set the labels for the x-and y-axes, convert S to character vectors. Use arrayfun to apply char to every element of S to return T. Set the XTickLabel and YTickLabel properties of a to T.

The solutions lie at the intersection of the two plots. This shows the system has repeated, periodic solutions. To solve this system of equations for the full solution set, use solve and set the ReturnConditions option to true.

solve returns a structure S with the fields S.x for the solution to x, S.y for the solution to y, S.parameters for the parameters in the solution, and S.conditions for the conditions on the solution. Elements of the same index in S.x, S.y, and S.conditions form a solution. Thus, S.x(1), S.y(1), and S.conditions(1) form one solution to the system of equations. The parameters in S.parameters can appear in all solutions.

You can use the solutions, parameters, and conditions returned by solve to find solutions within an interval or under additional conditions. This section has the same goal as the previous section, to solve the system of equations within a search range, but with a different approach. Instead of placing conditions directly, it shows how to work with the parameters and conditions returned by solve.

For the full solution S of the system of equations, find values of x and y in the interval -2*pi to 2*pi by solving the solutions S.x and S.y for the parameters S.parameters within that interval under the condition S.conditions.

Note that solx and soly are the two sets of solutions to x and to y. The full sets of solutions to the system of equations are the two sets of points formed by all possible combinations of the values in solx and soly.

Symbolic calculations provide exact accuracy, while numeric calculations are approximations. Despite this loss of accuracy, you might need to convert symbolic results to numeric approximations for use in numeric calculations. For a high-accuracy conversion, use variable-precision arithmetic provided by the vpa function. For standard accuracy and better performance, convert to double precision using double.

Y = solve(eqns,vars) solves the system of equations eqns for the variables vars and returns a structure that contains the solutions. If you do not specify vars, solve uses symvar to find the variables to solve for. In this case, the number of variables that symvar finds is equal to the number of equations eqns.

[y1,...,yN] = solve(eqns,vars) solves the system of equations eqns for the variables vars. The solutions are assigned to the variables y1,...,yN. If you do not specify the variables, solve uses symvar to find the variables to solve for. In this case, the number of variables that symvar finds is equal to the number of output arguments N.

[y1,...,yN,parameters,conditions]= solve(eqns,vars,'ReturnConditions',true) returns the additional arguments parameters and conditions that specify the parameters in the solution and the conditions on the solution.

When solving for more than one variable, the order in which you specify the variables defines the order in which the solver returns the solutions. Assign the solutions to variables solv and solu by specifying the variables explicitly. The solver returns an array of solutions for each variable.

By default, solve does not apply simplifications that are not valid for all values of x. In this case, the solver does not assume that x is a positive real number, so it does not apply the logarithmic identity log(3x)=log(3)+log(x). As a result, solve cannot solve the equation symbolically.

solve applies simplifications that allow the solver to find a solution. The mathematical rules applied when performing simplifications are not always valid in general. In this example, the solver applies logarithmic identities with the assumption that x is a positive real number. Therefore, the solutions found in this mode should be verified.

Try to get an explicit solution for such equations by calling the solver with 'MaxDegree'. The option specifies the maximum degree of polynomials for which the solver tries to return explicit solutions. The default value is 2. Increasing this value, you can get explicit solutions for higher order polynomials.

Equation to solve, specified as a symbolic expression or symbolic equation. The relation operator == defines symbolic equations. If eqn is a symbolic expression (without the right side), the solver assumes that the right side is 0, and solves the equation eqn == 0.

Maximum degree of polynomial equations for which solver uses explicit formulas, specified as a positive integer smaller than 5. The solver does not use explicit formulas that involve radicals when solving polynomial equations of a degree larger than the specified value.

Solutions of a system of equations, returned as a structure. The number of fields in the structure correspond to the number of independent variables in a system. If 'ReturnConditions' is set to true, the solve function returns two additional fields that contain the parameters in the solution, and the conditions under which the solution is true.

Solutions of a system of equations, returned as symbolic variables. The number of output variables or symbolic arrays must be equal to the number of independent variables in a system. If you explicitly specify independent variables vars, then the solver uses the same order to return the solutions. If you do not specify vars, the toolbox sorts independent variables alphabetically, and then assigns the solutions for these variables to the output variables.

Parameters in a solution, returned as a vector of generated parameters. This output argument is only returned if ReturnConditions is true. If a single output argument is provided, parameters is returned as a field of a structure. If multiple output arguments are provided, parameters is returned as the second-to-last output argument. The generated parameters do not appear in the MATLAB workspace. They must be accessed using parameters.

Conditions under which solutions are valid, returned as a vector of symbolic expressions. This output argument is only returned if ReturnConditions is true. If a single output argument is provided, conditions is returned as a field of a structure. If multiple output arguments are provided, conditions is returned as the last output argument.

If solve cannot find a solution and ReturnConditions is false, the solve function internally calls the numeric solver vpasolve that tries to find a numeric solution. For polynomial equations and systems without symbolic parameters, the numeric solver returns all solutions. For nonpolynomial equations and systems without symbolic parameters, the numeric solver returns only one solution (if a solution exists).

If the solution contains parameters and ReturnConditions is true, solve returns the parameters in the solution and the conditions under which the solutions are true. If ReturnConditions is false, the solve function either chooses values of the parameters and returns the corresponding results, or returns parameterized solutions without choosing particular values. In the latter case, solve also issues a warning indicating the values of parameters in the returned solutions.

Parameters introduced by solve do not appear in the MATLAB workspace. They must be accessed using the output argument that contains them. Alternatively, to use the parameters in the MATLAB workspace use syms to initialize the parameter. For example, if the parameter is k, use syms k.

The output variables y1,...,yN do not specify the variables for which solve solves equations or systems. If y1,...,yN are the variables that appear in eqns, then there is no guarantee that solve(eqns) will assign the solutions to y1,...,yN using the correct order. Thus, when you run [b,a] = solve(eqns), you might get the solutions for a assigned to b and vice versa.

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