Thiele's interpolation formula

Thiele's interpolation formula
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In mathematics, Thiele's interpolation formula is an interpolation formula for a function f(•) of a single variable, due to the Danish mathematician Thorvald Nicolai Thiele. It is expressed as a continued fraction, where ρ represents the reciprocal difference:

$f(x)=f(x_{1})+{\cfrac {x-x_{1}}{\rho (x_{1},x_{2})+{\cfrac {x-x_{2}}{\rho _{2}(x_{1},x_{2},x_{3})-f(x_{1})+{\cfrac {x-x_{3}}{\rho _{3}(x_{1},x_{2},x_{3},x_{4})-\rho (x_{1},x_{2})+\cdots }}}}}}$

Demonstrate Thiele's interpolation function by:

1. Building a 32 row trig table of values of the trig functions sin, cos and tan. e.g. for x from 0 by 0.05 to 1.55...
2. Using columns from this table define an inverse - using Thiele's interpolation - for each trig function;
3. Finally: demonstrate the following well known trigonometric identities:
   • 6 × sin^-1 ½ = π
   • 3 × cos^-1 ½ = π
   • 4 × tan^-1 1 = π

ALGOL 68

Works with: ALGOL 68 version Revision 1 - except the Currying (aka partial parametrisation) in test block is a proposal for ALGOL 68 Rev2

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny - Currying is supported.

<lang algol68>PROC raise exception = ([]STRING msg)VOID: ( putf(stand error,("Exception:", $" "g$, msg, $l$)); stop );

1. The MODE of lx and ly here should really be a UNION of "something REAL",

"something COMPLex", and "something SYMBOLIC" ... #

PROC thiele=([]REAL lx,ly, REAL x) REAL: BEGIN

 []REAL xx=lx[@1],yy=ly[@1];
 INT n=UPB xx;
 IF UPB yy=n THEN

1. Assuming that the values of xx are distinct ... #

   [0:n-1,1:n]REAL p;
   p[0,]:=yy[];
   FOR i TO n-1 DO p[1,i]:=(xx[i]-xx[1+i])/(p[0,i]-p[0,1+i]) OD;
   FOR i FROM 2 TO n-1 DO
     FOR j TO n-i DO
       p[i,j]:=(xx[j]-xx[j+i])/(p[i-1,j]-p[i-1,j+1])+p[i-2,j+1]
     OD
   OD;
   REAL a:=0;
   FOR i FROM n-1 BY -1 TO 2 DO a:=(x-xx[i])/(p[i,1]-p[i-2,1]+a) OD;
   yy[1]+(x-xx[1])/(p[1,1]+a)
 ELSE
   raise exception(("Unequal length arrays supplied: ",whole(UPB xx,0)," NE ",whole(UPB yy,0))); SKIP
 FI

END;

test:(

 FORMAT real fmt = $g(0,real width-2)$;

 REAL lwb x=0, upb x=1.55, delta x = 0.05; 

 [0:ENTIER ((upb x-lwb x)/delta x)]STRUCT(REAL x, sin x, cos x, tan x) trig table;

 PROC init trig table = VOID: 
   FOR i FROM LWB trig table TO UPB trig table DO 
     REAL x = lwb x+i*delta x; 
     trig table[i]:=(x, sin(x), cos(x), tan(x))
   OD;

 init trig table;

1. Curry the thiele function to create matching inverse trigonometric functions #

 PROC (REAL)REAL inv sin = thiele(sin x OF trig table, x OF trig table,),
                 inv cos = thiele(cos x OF trig table, x OF trig table,),
                 inv tan = thiele(tan x OF trig table, x OF trig table,);

 printf(($"pi estimate using "g" interpolation: "f(real fmt)l$, 
   "sin", 6*inv sin(1/2),
   "cos", 3*inv cos(1/2),
   "tan", 4*inv tan(1)
 ))

)</lang> Output:

pi estimate using sin interpolation: 3.1415926535898
pi estimate using cos interpolation: 3.1415926535898
pi estimate using tan interpolation: 3.1415926535898