Differences

This shows you the differences between two versions of the page.

--- playground:playground [2021/09/29 15:20]
jojo1973
+++ playground:playground [2021/10/01 09:08]
jojo1973 removed
@@ Line 1: / Line 1: @@
-==== Numerical integration ====
+===== Numerical integration =====
 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]].
@@ Line 39: / Line 39: @@
 ----
-=== Example 1: Bound function on closed interval ===
+==== Angles and trigonometric expressions ====
-|  pic  | $$ \int^{2}_{0} x^{10}e^{\left(4x^{3}-3x^{4}\right)} \,dx $$  |||
+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)'
+
+
+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  | **7.258 395 170 614 291** ...  |   Δ ≤ 10<sup>-16</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 ====
+----
+==== Example 4: XXX ====
-|{{: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! |