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playground:playground [2021/09/27 13:26]
jojo1973 		playground:playground [2021/10/01 09:08]
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-==== Numerical integration ====+===== Numerical integration =====
  
-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]].+Numeric integration of symbolic expressions is performed 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: ''[[manual:chapter6:solvers:cmd_numint|NUMINT]]'' accepts four arguments:
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 For example valid expressions are: For example valid expressions are:
 +
 <code> <code>
-F(X)=X*LN(X) +'F(X)=X*LN(X)' 
-G(Z)=Z^2-2*COS(Z) +'G(Z)=Z^2-2*COS(Z)' 
-VEL(T)=ACC*T+'VEL(T)=ACC*T'
 </code> </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. 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.
  
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 ---- ----
  
-=== Examples ===+==== Angles and trigonometric expressions ====
  
-The following examples are calculated with 16 precision digits using three different tolerances. If available, the analytic result is provided for comparison.+When a real number is input to a trigonometric function **newRPL** assumes that it is an angle expressed in the current angular mode; however the trigonometric functions are meant to process quantities expressed in radians. In other words the following transformations are implicitly applied: 
 + 
 +  * in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode $ θ=θ_{r}\, $; 
 +  * in ''[[manual:chapter6:flags:cmd_deg|DEG]]'' mode $ θ=\frac{\pi}{180}{θ_°}\, $; 
 +  * in ''[[manual:chapter6:flags:cmd_grad|GRAD]]'' mode $ θ=\frac{\pi}{200}{θ_g}\, $. 
 + 
 +where $θ_{r}\,$, $θ_{°}$ and $θ_{g}$ are the quantities entered by the user and $θ$ is what is actually fed to the trigonometric functions. 
 + 
 +From this, two important consequences derive: 
 + 
 +  - the antiderivative changes according to the angular mode. Let's consider e.g. the function $ f(θ)=\sin{θ} $ and apply the transformations above:\\ \\ 
 +    * in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode $ f(θ_r) $ is equivalent to $ f(θ) $ which differentiates to $ f(θ_r)\,dθ_r=\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\cos{θ}\, $; 
 +    * in ''[[manual:chapter6:flags:cmd_deg|DEG]]'' mode $ f(θ_°) $ is actually $ f\left(\frac{\pi}{180}θ_°\right) $ which differentiates to $ f\left(\frac{\pi}{180}θ_°\right)\,dθ_°=\frac{180}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\frac{180}{\pi}\cos{θ}\, $; 
 +    * in ''[[manual:chapter6:flags:cmd_grad|GRAD]]'' mode $ f(θ_g) $ is actually $ f\left(\frac{\pi}{200}θ_g\right) $ which differentiates to $ f\left(\frac{\pi}{200}θ_g\right)\,dθ_g=\frac{200}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\frac{200}{\pi}\cos{θ}\, $;\\ \\ 
 +  - no inverse transformation is applied to the resulting output: this is mathematically correct, but can be disconcerting if one is not immediately aware of the implicit variable substitution. 
 +\\ 
 +In conclusion, unless the user knows exactly what he/she is doing it's advisable **to perform numeric integration of trigonometric expressions in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode**. 
 + 
 +---- 
 + 
 +==== Example 1: Bound function on closed interval ==== 
 + 
 +|  pic  |  $$ \int^{2}_{0} x^{10}e^{\left(4x^{3}-3x^{4}\right)} \,dx $$  | **Input:**\\ <code>16 SETPREC 
 +'F(X)=X^10*EXP(4*X^3-3*X^4)' 
 +0 
 +2 
 +tol 
 +NUMINT</code>  || 
 +|  :::  |  tol=10<sup>-4</sup>   | **7.258 3**76 114 514 225 ...  |  Δ = -1.9·10<sup>-5</sup>  | 
 +|  :::  |  tol=10<sup>-8</sup>   | **7.258 395 17**3 115 920 ...  |  Δ = 2.5·10<sup>-9</sup>  | 
 +|  :::  |  tol=10<sup>-12</sup>  | **7.258 395 170 61**5 141 ...  |  Δ = 8.5·10<sup>-13</sup>  | 
 +|  :::  |  Exact at the precision shown  | **[[https://www.wolframalpha.com/input/?i=integrate+x%5E10*exp%284*x%5E3-3*x%5E4%29+from+0+to+2+to+16+digits|7.258 395 170 614 291]]** ...  |   Δ ≤ 10<sup>-16</sup>  | 
 + 
 +---- 
 + 
 +==== Example 2: XXX ==== 
 + 
 +---- 
 + 
 +==== Example 3: XXX ==== 
 + 
 +----
  
-^  Function  ^  Tolerance  ^^^  Exact Solution  ^^ +==== Example 4: XXX ====
-^    ^  ''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! | 
  • playground/playground.txt
  • Last modified: 2021/10/11 13:50
  • by jojo1973