Hello,<\/p>\n
Today I will present how flexible mXparser really is. As an example we will approximate $$\\sin(x)$$ and\u00a0$$\\cos(x)$$ using indirect recursion<\/strong><\/em><\/span> steps, which means two functions depending on each other. Let us start with a bit of theory starting with basic\u00a0trigonometric identities<\/a><\/strong>:<\/p>\n $$\\sin(2x)=2\\sin(x)\\cos(x)$$ Above formals can be equivalently written as<\/p>\n $$\\sin(x)=2\\sin\\big(\\frac{x}{2}\\big)\\cos\\big(\\frac{x}{2}\\big)$$ Please notice that knowing the solution for smaller value $$\\frac{x}{2}$$ it is possible to get solution for the original\u00a0one $$x$$. \u00a0This simply means that mentioned trigonometric identities are in fact example of recursion<\/strong> <\/em><\/span>– here indirect recursion as $$\\sin(x)$$ function is using $$\\cos(x)$$, and $$\\cos(x)$$ is using $$\\sin(x)$$. The complete recursion definition requires base case definition<\/span><\/strong><\/em><\/span> (stop condition).<\/p>\n For $$x$$ near to $$0$$ function $$\\sin(x)$$ can be well approximated exactly by $$x$$, while in case of\u00a0$$\\cos(x)$$ constant $$1$$ is pretty good approximation. This gives good stop condition.<\/p>\n For\u00a0small $$a>0$$ let us define two recursive functions:<\/p>\n $$\\text{s}(x)=\\begin{cases}x&\\text{dla}\\quad |x|<a\\\\2\\text{s}\\big(\\frac{x}{2}\\big)\\text{c}\\big(\\frac{x}{2}\\big)&\\text{dla}\\quad |x|\\geq a\\end{cases}$$ Once again please notice that $$\\text{s}$$ (and $$\\text{c}$$ separately) is using both $$\\text{s}\\big(\\frac{x}{2}\\big)$$ and $$\\text{c}\\big(\\frac{x}{2}\\big)$$.<\/p>\n We expect that smaller parameter $$a$$ is giving better approximations – below you will find functions graphs separately for $$a = 0.5$$ and $$a=0.01$$.<\/p>\n
\n$$\\cos(2x)=\\cos^2(x)-\\sin^2(x)$$<\/p>\n
\n$$\\cos(x)=\\cos^2\\big(\\frac{x}{2}\\big)-\\sin^2\\big(\\frac{x}{2}\\big)$$<\/p>\n
\n$$\\text{c}(x)=\\begin{cases}1&\\text{dla}\\quad |x|<a\\\\\\text{c}^2\\big(\\frac{x}{2}\\big)-\\text{s}^2\\big(\\frac{x}{2}\\big)&\\text{dla}\\quad |x|\\geq a\\end{cases}$$<\/p>\n\r\nimport org.mariuszgromada.math.mxparser.*;\r\n...\r\n\/* Recursive functions definition *\/\r\nConstant a = new Constant("a", 0.1);\r\nFunction s = new Function("s(x) =\u00a0 if( abs(x) < a, x, 2*s(x\/2)*c(x\/2) )", a);\r\nFunction c = new Function("c(x) =\u00a0 if( abs(x) < a, 1, c(x\/2)^2-s(x\/2)^2 )", a);\r\n\r\n\/* Pointing that 's' is using 'c', and 'c' is using 's' *\/\r\ns.addDefinitions(c);\r\nc.addDefinitions(s);\r\n\r\n<\/pre>\n