How to resolve the algorithm Continued fraction step by step in the Maxima programming language

Published on 12 May 2024 09:40 PM
#Maxima

How to resolve the algorithm Continued fraction step by step in the Maxima programming language

Table of Contents

Problem Statement

The task is to write a program which generates such a number and prints a real representation of it. The code should be tested by calculating and printing the square root of 2, Napier's Constant, and Pi, using the following coefficients: For the square root of 2, use

a

0

= 1

{\displaystyle a_{0}=1}

then

a

N

= 2

{\displaystyle a_{N}=2}

.

b

N

{\displaystyle b_{N}}

is always

1

{\displaystyle 1}

. For Napier's Constant, use

a

0

= 2

{\displaystyle a_{0}=2}

, then

a

N

= N

{\displaystyle a_{N}=N}

.

b

1

= 1

{\displaystyle b_{1}=1}

then

b

N

= N − 1

{\displaystyle b_{N}=N-1}

. For Pi, use

a

0

= 3

{\displaystyle a_{0}=3}

then

a

N

= 6

{\displaystyle a_{N}=6}

.

b

N

= ( 2 N − 1

)

2

{\displaystyle b_{N}=(2N-1)^{2}}

.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Continued fraction step by step in the Maxima programming language

Source code in the maxima programming language

cfeval(x) := block([a, b, n, z], a: x[1], b: x[2], n: length(a), z: 0,
   for i from n step -1 thru 2 do z: b[i]/(a[i] + z), a[1] + z)$

cf_sqrt2(n) := [cons(1, makelist(2, i, 2, n)), cons(0, makelist(1, i, 2, n))]$

cf_e(n) := [cons(2, makelist(i, i, 1, n - 1)), append([0, 1], makelist(i, i, 1, n - 2))]$

cf_pi(n) := [cons(3, makelist(6, i, 2, n)), cons(0, makelist((2*i - 1)^2, i, 1, n - 1))]$

cfeval(cf_sqrt2(20)), numer;   /* 1.414213562373097 */
% - sqrt(2), numer;            /* 1.3322676295501878*10^-15 */

cfeval(cf_e(20)), numer;       /* 2.718281828459046 */
% - %e, numer;                 /* 4.4408920985006262*10^-16 */

cfeval(cf_pi(20)), numer;      /* 3.141623806667839 */
% - %pi, numer;                /* 3.115307804568701*10^-5 */


/* convergence is much slower for pi */
fpprec: 20$
x: cfeval(cf_pi(10000))$
bfloat(x - %pi);               /* 2.4999999900104930006b-13 */

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