什么是 closed form solution

Closed form solution 是指可以用一个公式或者表达式来直接求得解的问题。这个公式或者表达式可以包含基本的数学运算、常量和变量，但不能包含未知的函数或者变量。简单来说，closed form solution 就是能够用一个封闭的数学表达式来计算出问题的解，而不需要进行迭代或者逼近的计算。在某些数学问题中，存在closed form solution 可以大大简化问题的求解过程。

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closed form solution

A closed form solution refers to a mathematical expression that can be evaluated in a finite number of operations. In other words, a closed form solution provides an explicit formula for computing the solution to a mathematical problem without the need for iterative or numerical methods.

For example, the quadratic formula is a closed form solution that provides the roots of a quadratic equation. Similarly, the formula for the sum of an arithmetic sequence is a closed form solution that allows us to calculate the sum directly, without the need for adding up each term in the sequence.

However, not all mathematical problems have closed form solutions. In some cases, numerical methods or approximation techniques may be necessary to obtain an answer.

matlab unable to solve symbolically. returning a numeric solution using vpasolve.

In MATLAB, if the symbolic solver is unable to find a closed-form solution for an equation, you can use the `vpasolve` function to obtain a numerical solution instead. `vpasolve` stands for "Variable Precision Arithmetic Solve" and it can handle equations involving both symbolic and numeric variables.

To use `vpasolve`, you need to define your equation as a symbolic expression using the `syms` function, specify the variables you want to solve for, and then call `vpasolve` with the equation and initial guesses for the variables. Here's an example:

syms x y
eqn = x^2 + y^2 == 1;
sol = vpasolve(eqn, [x, y], [0.5, 0.5]);

In this example, `eqn` represents the equation x^2 + y^2 = 1. The `vpasolve` function is called with the equation, the variables to solve for (x and y), and initial guesses for the variables ([0.5, 0.5]). The `sol` variable will store the numerical solutions.

Keep in mind that `vpasolve` can be slower than the symbolic solver, especially for complex equations or large systems of equations. It is also worth noting that the numerical solutions obtained using `vpasolve` might have limited precision compared to exact symbolic solutions.