approximation-methods

Approximation Methods for Equations

Summary: Two iterative methods for finding roots numerically when no exact algebraic formula is available: a Newton-like linearization (Method 1) and a recurrence-series ratio method (Method 2), both described by Euler in Chapter XVI.

Sources: chapter-1.4.16, additions-1

Last updated: 2026-05-10

---

Context

When equations exceed degree 4, or when the roots of lower-degree equations are irrational, exact algebraic solutions may be unwieldy or non-existent (source: chapter-1.4.16, §784). Approximation gives a sequence of rational numbers converging to a root.

Method 1: Newton-like Linearization

Given an initial estimate $n$ close to a root, write $x=n+p$ (or $x=n-p$). Substitute into the equation and drop all powers of $p$ above the first; solve the resulting linear equation for $p$.

The corrected value $x=n+p$ is a better approximation, and can be used as a new $n$ for the next iteration.

Formulas for pure power equations

Equation	Iterative formula
$x^2=a$	$x\leftarrow \frac{n^2+a}{2n}$
$x^3=a$	$x\leftarrow \frac{2n^3+a}{3n^2}$

These are classical Babylonian / Heron formulas. Both converge very rapidly (roughly squaring the number of correct digits per step).

General polynomial

For $x^3+a{x}^2+bx+c=0$ the formula is (source: §789):

$x\leftarrow \frac{2n^3+a{n}^2-c}{3n^2+2an+b}$

For degree $k$, the denominator is the derivative of the polynomial evaluated at $n$ — this is Newton’s method, though Euler does not name it as such.

Applicability

Works for all degrees and all equation types. The main requirement is a reasonably close initial estimate. If the initial guess is too far off (as in §791 with $n=1$ for $x^5=6x+10$), the correction can overshoot; choosing $n=2$ recovers accuracy.

Method 2: Recurrence Series

Build a sequence $a_0,a_1,a_2,\dots$ whose recurrence is derived from the equation. The ratio $a_{n+1}/a_n$ converges to the largest real root of the equation.

Deriving the recurrence

For the equation $x^m=c_1{x}^{m-1}+c_2{x}^{m-2}+\cdots +c_m$, set $x^k=s_{n+k}/s_n$ for all $k$. Substituting gives:

$s_{n+m}=c_1{s}_{n+m-1}+c_2{s}_{n+m-2}+\cdots +c_m{s}_n$

A linear recurrence in $s_n$ with the same coefficients as the equation.

Key examples

Equation	Recurrence	Series (starting 0,1)	Root
$x^2=x+1$	$s_{n+2}=s_{n+1}+s_n$	$0,1,1,2,3,5,8,13,\dots$	$\varphi =\frac{1+\sqrt{5}}{2}$
$x^2=2x+1$	$s_{n+2}=2s_{n+1}+s_n$	$0,1,2,5,12,29,70,\dots$	$1+\sqrt{2}$

The Fibonacci sequence arises naturally as the approximation sequence for the golden ratio.

Convergence to largest root

The series always converges to the greatest root. For the smaller root, the initial terms must be chosen to match that root exactly, otherwise the series drifts toward the larger root (source: §799).

Limitation: missing second term

When the equation lacks an $x^{m-1}$ term (e.g., $x^2=2$), the recurrence has zero coefficient for the preceding term, and consecutive ratios oscillate between two values rather than converging. Fix: substitute $x=y-1$ (or another shift) to introduce a linear term before building the series.

Infinite-degree equation (§800)

Euler notes that $x^{\infty }=x^{\infty -1}+x^{\infty -2}+\cdots$ requires each series term to equal the sum of all predecessors, forcing the series $1,1,2,4,8,16,\dots$. This gives root $x=2$ (also confirmed analytically by summing the geometric series).

Comparison

Euler concludes (§801) that Method 1 deserves preference for its universality. Method 2 is elegant but requires the equation to be in a compatible form, fails when the second term is missing, and only converges to the largest root automatically.

Method 3: Continued-Fraction Root Extraction (Lagrange, Add. I, Art. 9)

Lagrange supplies a third approach in his first Addition. Given an equation in $x$, find the integer $\alpha$ nearest a root and substitute $x=\alpha +1/y$ — the resulting equation in $y$ has a root $>1$. Find its integer part $\beta$, substitute $y=\beta +1/z$, and so on. The root is then expressed as

$x=\alpha +\frac{1}{\beta +\frac{1}{\gamma +\cdots }}.$

The CF terminates iff the root is rational; otherwise it goes on indefinitely, with each truncation a best rational approximation to the root.

This method has two advantages over Methods 1–2:

1. It detects rationality automatically (terminating expansion ⇔ rational root).
2. The successive truncations have provable error bounds $1/q_n^2$ (see convergents), which Newton’s method does not provide a priori.

See continued-fractions and add1-continued-fractions for the full theory; the connection to roots of quadratics is developed further in subsequent Additions.

Related pages

• ch1.4.16-approximation
• quartic-equations
• cubic-equations
• quadratic-equations
• geometrical-progressions
• infinite-series
• equations
• continued-fractions
• convergents
• add1-continued-fractions