Continued fraction/Arithmetic

By popular demand, see Talk:Continued fraction#creating_a_continued_fraction and Talk:Continued fraction#Arithmetics.3F.3F, or be careful what you ask for.

This page is a placeholder for several subtasks which will eventually implement a function:

G(matrix NG, Continued Fraction N₁, Continued Fraction N₂)

which will perform basic mathmatical operations on continued fractions.

Mathworld informs me:

Gosper has invented an algorithm for performing analytic addition, subtraction, multiplication, and division using continued fractions. It requires keeping track of eight integers which are conceptually arranged at the polyhedron vertices of a cube. Although this algorithm has not appeared in print, similar algorithms have been constructed by Vuillemin (1987) and Liardet and Stambul (1998).

Gosper's algorithm for computing the continued fraction for (ax+b)/(cx+d) from the continued fraction for x is described by Gosper (1972), Knuth (1998, Exercise 4.5.3.15, pp. 360 and 601), and Fowler (1999). (In line 9 of Knuth's solution, X_k<-|_A/C_| should be replaced by X_k<-min(|_A/C_|,|_(A+B)/(C+D)_|).) Gosper (1972) and Knuth (1981) also mention the bivariate case (axy+bx+cy+d)/(Axy+Bx+Cy+D).

My description follows part of Gosper reproduced on perl.plover.com. This document is text and unnumbered, you may wish to start by searching for "Addition, Multiplication, etc. of Two Continued Fractions" prior to reading the whole thing.

perl.plover.com also includes a series of slides as a class on continued fractions. The example [1;5,2] + 1/2 in G(matrix NG, Contined Fraction_N) is worked through in this class.

For these tasks continued fractions will be of the form:

$a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\ddots }}}}}}$

so each may be described by the notation [$a_{0};a_{1},a_{2},...,a_{n}$]

Continued fraction/Arithmetic/Construct from rational number

During these tasks I shall use the function described in this task to create continued fractions from rational numbers.

Matrix NG

Consider a matrix NG:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$

and a function $G(\mathrm {matrix}$ ${\mathit {NG}},\mathrm {Number}$ $N_{1},\mathrm {Number}$ $N_{2})$ which returns:

${\frac {a_{12}\times N_{1}\times N_{2}+a_{1}\times N_{1}+a_{2}\times N_{2}+a}{b_{12}\times N_{1}\times N_{2}+b_{1}\times N_{1}+b_{2}\times N_{2}+b}}$

Basic identities

${\mathit {NG}}={\begin{bmatrix}0&1&1&0\\0&0&0&1\end{bmatrix}}$

computes $N_{1}+N_{2}$

${\mathit {NG}}={\begin{bmatrix}0&1&-1&0\\0&0&0&1\end{bmatrix}}$

computes $N_{1}-N_{2}$

${\mathit {NG}}={\begin{bmatrix}1&0&0&0\\0&0&0&1\end{bmatrix}}$

computes $N_{1}\times N_{2}$

${\mathit {NG}}={\begin{bmatrix}0&1&0&0\\0&0&1&0\end{bmatrix}}$

computes $N_{1}/N_{2}$

Compound operations

${\mathit {NG}}={\begin{bmatrix}21&-15&28&-20\\0&0&0&1\end{bmatrix}}$

calculates ($3\times N_{1}+4)\times (7\times N_{2}-5)$

Note that with $N_{1}=22$, $N_{2}=7$, and

${\mathit {NG}}={\begin{bmatrix}0&1&0&0\\0&0&1&0\end{bmatrix}}$

I could define the solution to be $N_{1}=1$, $N_{2}=1$ and

${\mathit {NG}}={\begin{bmatrix}0&0&0&22\\0&0&0&7\end{bmatrix}}$

So I can define arithmetic as operations on this matrix which make $a_{12}$, $a_{1}$, $a_{2}$, $b_{12}$, $b_{1}$, $b_{2}$ zero and read the answer from $a$ and $b$. This is more interesting when $N_{1}$ and $N_{2}$ are continued fractions, which is the subject of the following tasks.

G(matrix NG, Contined Fraction N)

Here we perform basic mathematical operations on a single continued fraction.

The bivariate solution G(matrix NG, Continued Fraction N₁, Continued Fraction N₂)

Here we perform basic mathematical operations on two continued fractions.

• Compare two continued fractions
• Sqrt of a continued fraction