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playground:playground [2021/09/27 13:26]
jojo1973 		playground:playground [2021/10/11 13:50] (current)
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-==== Numerical integration ==== +Have fun!
- +
-Numeric integration of symbolic expressions is implemented in **newRPL** via the ''[[manual:chapter6:solvers:cmd_numint|NUMINT]]'' command which implements the [[https://en.wikipedia.org/wiki/Adaptive_Simpson%27s_method|Adaptive Simpson's method]]. +
- +
-''[[manual:chapter6:solvers:cmd_numint|NUMINT]]'' accepts four arguments: +
- +
-  * the mono-variate function to integrate, either in __symbolic__ or __program__ form; +
-  * the lower integration limit; +
-  * the upper integration limit; +
-  * the error tolerance. +
- +
-If the **function** to integrate is expressed in symbolic form it must respect a precise syntax: +
- +
-  * it must be written as an equation; +
-  * the left side must be in the form ''//func//(//var//)'' where ''//var//'' is the integrating variable and ''//func//'' is the function's name; +
-  * the right side is a function, expressed in terms of ''//var//''. +
- +
-For example valid expressions are: +
-<code> +
-F(X)=X*LN(X) +
-G(Z)=Z^2-2*COS(Z) +
-VEL(T)=ACC*T +
-</code> +
-As shown above, the functions may refer to global or local variables; the function's name is only descriptive and bears no relevance to the calculation. +
- +
-Alternatively, the function can be written as a program which __accepts exactly one numeric argument and returns exactly one numeric result__. The expressions above can be rewritten as: +
-<code> +
-« DUP LN * » +
-« DUP SQ SWAP COS 2 * - » +
-« 'ACC' RCL * » +
-</code> +
- +
-The **integration limits** can be either real or complex __finite__ numbers; symbolic constants are accepted and silently converted to numerical values. +
- +
-The **error tolerance** is a real number used to specify the required precision of the calculation: when two successive iterations differ by a value which is less than the tolerance the calculation stops. +
- +
----- +
- +
-=== Examples === +
- +
-The following examples are calculated with 16 precision digits using three different tolerances. If available, the analytic result is provided for comparison. +
- +
-^  Function  ^  Tolerance  ^^^  Exact Solution  ^^ +
-^    ^  ''1E-4''  ^  ''1E-8''  ^  ''1E-12''  ^    ^ +
-| $ \int^{5/4}_{0} \cos x^2 \,dx $  |  0.977 4  |  0.977 437 67  |  0.977 437 670 720  |  $ \sqrt\frac{\pi}{2} C\left(\sqrt\frac{2}{\pi} x\right) \Biggr|^{x=5/4}_{x=0} $  |  0.977 437 670 720 3...  | +
-| $ \int^{2}_{0} e^{-2x}\left(14x-11x^{2}\right) \,dx $  |  1.084 2  |  1.084 260 41  |  1.084 260 409 719  |  $  \frac{3}{4}+\frac{73}{4e^{4}} $  |  1.084 260 409 719 3...  | +
-| $ \int^{2}_{0} x^{10}e^{\left(4x^{3}-3x^{4}\right)} \,dx $  |  7.258 4  |  7.258 395 17  |  7.258 395 170 615  |  No analytic solution exists  |  7.258 395 170 614 3...  | +
- +
-|{{:manual:warning-145066_640.png?100 |Warning!}} The user must take care of implicit substitutions when the expression to integrate __contains trigonometric functions__. For example, if the angle mode is set to degrees, the expression $ \sin x $ is actually interpreted as $ \sin\left(\frac{\pi}{180}x\right) $, whose anti-derivative is **not** $-\cos x$, but $-\frac{180}{\pi}\cos\left(\frac{\pi}{180}x\right) $. Unexpected results will follow if the user disregards this occurrence! |+
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  • Last modified: 2021/09/27 13:26
  • by jojo1973