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Functions names are as follows: `nm-[operation type]-[method]`
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Creates the following functions:
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```typ
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#nm-differentiate-forward()
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#nm-differentiate-central()
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#nm-integrate-midpoint()
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#nm-integrate-trapezium()
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#nm-integrate-simpsons()
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#nm-iterate-FPI()
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#nm-iterate-relaxed-FPI()
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#nm-iterate-newton-raphson()
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#nm-iterate-secant()
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#nm-iterate-false-position()
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#nm-iterate-bisection()
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#nm-table-integrate()
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#nm-table-differentiate()
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#nm-table-iterate()
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#plot-integral()
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```
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## nm-differentiation
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Differentiation tools take the following inputs/defaults:
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```typ
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f_x:x=>x*x, x0:1, h:1, accuracy:12
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```
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The function will approximate $f'(x_0)$ using either the Forward Difference or Central Difference method.
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## nm-integrate
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Integration tools take the following inputs/defaults:
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```typ
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f_x:x=>x*x, x0:0, x1:1, accuracy:12, n:1
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```
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These will approximate $\int_{x 0}^{x 1}(f_x(x))\text{d}x $
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Trapezium rule will return $T_n$, Midpoint rule will return $M_n$ and Simpsons rule will return $S_{2n}$
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## nm-iterate
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These functions provide the root-finding methods Bisection, False Position, Secant, Newton Raphson, Fixed Point Iteration (FPI) and Relaxed Fixed Point Iteration.
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All functions take the parameter `f_x`, except FPI and Relaxed FPI, which take the function `g_x`.
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Bisection and False Position will return the region in which the root lies, whereas the others will return their nearest approximation.
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The parameter `return-all` can be made true to see every step of the iteration, rather than just the last one requested.
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*NOTE*: The Newton-Raphson method utilises the Central Difference gradient with `h = 0.0000000001` calculated to 15 decimal places. Thus it is not a true Newton-Raphson approximation.
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## nm-table
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These tables are useful for seeing or demonstrating convergence. They have been designed with the A Level Further Maths OCR MEI B specification in mind.
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By default, they will include the changing variable ($n$ or $h$ typically), as well as the approximation they have reached.
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Optionally, the differences between the estimates can be shown.
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If the differences are being shown, then the ratios between the differences can also be shown, with a customizeable order for iterative methods.
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*NOTE*: For exam purposes, differences and ratios are caluclated from the *table values*, which are rounded, rather than the greater precision stored values.
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## plot-integral
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Useful for demonstrating different types of numerical integration methods. Uses CeTZ to plot.
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The number of strips is customizable, and the method should be chosen from:
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`integral,mid,left,right,trapezium`
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If `integral` is chosen, or any non-listed input is given for the `method` variable, the function defaults to simply highlighting the area to be found.
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*NOTE*: Labels aren't in mathematical format, simply New Computer Modern Math font.
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