func
©Convert vector v to polar
[R▶Pr(v[1,1],v[1,2]),R▶Pθ(v[1,1],v[1,2])]
EndFunc
rect(v)
func
©Convert vector v to rectangular
[P▶Rx(v[1,1],v[1,2]),P▶Ry(v[1,1],v[1,2])]
EndFunc
polar([1,1])
returns [[√(2),Œ/4]]. rect([[√(2),Œ/4]]) returns [1,1]. Note that these routines
assume the current Angle Mode setting. If the angle mode is radians, then the angles are in radians. If
you or your application expect the arguments or results to be in degrees or radians, set the proper
mode.
(Credit to Sam Jordan for prompting my write-up; code (and bugs) are mine)
[6.26] Accurate numerical derivatives with nDeriv() and Ridder's method
This tip is lengthy. It is divided into these sections:
! Optimum results from nDeriv. How to use the built-in nDeriv() function to get the most accurate
results, with a function called nder2().
! More accurate results with Ridders' method. Shows a method (Ridders') that gives much better
results than nDeriv(), with a program called nder1().
! Diagnosing nder1() with nder1p(). Shows how to fix errors that might occur with the Ridders'
method program.
! General comments on numerical differentiation. General concerns with any numerical differentiation
method.
! More comments on nder1() and Ridders' method. Lengthy discussion of nder1() results,
performance, and also some interesting behavior of the built-in nDeriv() function.
In general, the most accurate method to find a numerical derivative is to use the CAS to find the
symbolic derivative, then evaluate the derivative at the point of interest. For example, to find the
numeric derivative of tan(x), where x = /2.01 = 1.562981..., find
✜
d
dx
tan(x)=
1
(
cos
(
x
))
2
= 16374.241898666
This method fails when the CAS cannot find a symbolic expression for the derivative, for example, for a
complicated user function. The 89/92+ provide two built-in functions for finding numerical derivatives:
avgRC() and nDeriv(). avgRC() uses the forward difference to approximate the derivative at x:
avgRC(f(x),x,h)=
f(x+h)−f(x)
h
and nDeriv() uses the central difference to approximate the derivative:
nDeriv(f(x),x,h)=
f(x+h)−f(x−h)
2h
6 - 27
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