closed form solution

A closed-form solution is a mathematical expression that can be evaluated in a finite number of operations, using only known and accepted operations. In other words, it is a formula that provides an exact solution to a problem, without the need for iterative or numerical methods. Examples of problems that have closed-form solutions include linear equations, quadratic equations, and some integrals. The advantage of having a closed-form solution is that it can be computed quickly and accurately, without the need for approximations or simulations.

相关问题

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.

function [v_final]=Closed_form_solution_neuron(neuron_number,v_pre,delta_t,y_init,g_l,v_l,c_m,sigma,mu,W,E) y_1 = y_init; %[v_pre_resampled time_ref]= resample(v_pre,[1:size(v_pre,1)],1/delta_t); time = 0:delta_t:(size(v_pre{neuron_number},1)*delta_t)-delta_t; time = time'; v_connections_pre = zeros(size(v_pre{neuron_number},1),1); %tau = (y_1 - sum(E)).*exp(-time'.*[(g_l(1,1)./c_m(1,1)) + sum([W.*1./(exp(-sigma.*(v_pre{neuron_number}'-mu))+1)],1)./c_m(1,1)]); %v_final = tau.*sum(1./(exp(sigma.*(v_pre{neuron_number}'-mu))+1).^W,1)+sum(v_l(1,1)+E); for i=1:size(v_pre{neuron_number},2) v_connections(:,i) = (y_1 - E(i,1)).* exp(-time.*[(g_l(1,1)./c_m(1,1)) + [W(i)*1./(exp(-sigma(i,1).*(v_pre{neuron_number}(:,i)-mu(i,1)))+1)]/c_m(1,1)]).*(1./(exp(sigma(i,1).*(v_pre{neuron_number}(:,i)-mu(i,1)))+1).^W(i))+E(i); %v_connections(:,i) = (y_1 - E(i,1)).* exp(-time.*[(g_l(1,1)./c_m(1,1))]).*(1./(exp(sigma(i,1).*(v_pre{neuron_number}(:,i)-mu(i,1)))+1).^W(i))+E(i); v_final = v_connections_pre + v_connections(:,i); v_connections_pre = v_final; end %v_final = v_final + v_l(1,1); clear g_l v_l c_m sigma mu W E time v_connections

这段代码实现了一个神经元模型的闭合解析解法，输入参数包括预先记录的外部电压v_pre，时间间隔delta_t，初始状态y_init，以及神经元模型的参数g_l, v_l, c_m, sigma, mu, W和E。其中，g_l是静息电导，v_l是静息电位，c_m是膜电容，sigma和mu是激活函数的参数，W是突触权值矩阵，E是突触电位。该代码通过解析求解微分方程，计算出神经元的动态响应v_final，并返回结果。其中，v_connections表示每个突触的动态响应，v_connections_pre表示上一个突触的动态响应，通过循环累加每个突触的动态响应得到最终结果v_final。