Chapter 17 — Using Recurrent Series to Find Roots of Equations

Summary: §332–§355. Euler inverts the direction of Chapter 13: there, the roots of the denominator were known and the closed-form general term was extracted; here, only the coefficients of an equation $x^m-\alpha x^{m-1}-\beta x^{m-2}-\cdots =0$ are given, and the recurrent series is run in order to discover the roots. Following Daniel Bernoulli (Volume III of the Commentaries of the St. Petersburg Academy), the quotient of consecutive coefficients $Q/P$ of the recurrent series with scale $\alpha ,\beta ,\gamma ,\dots$ approaches the largest root in absolute value — the Bernoulli method. The chapter then catalogues the failure modes (roots close in size, $\pm p$ pairs, repeated roots, dominant complex pairs) and the remedies (the substitution $x=y+k$, alternate-ratio reading, numerator $=1$ as a safety, and the §348–§352 trinomial-factor extraction that recovers both modulus and argument of a dominant complex conjugate pair).

Sources: chapter17

Last updated: 2026-05-11

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Movement 1 — The basic correspondence (§332–§338)

A rational function with constant term $1$ in the denominator,

$\frac{a+bz+c{z}^2+d{z}^3+\cdots }{1-\alpha z-\beta z^2-\gamma z^3-\cdots },$

expands as a recurrent series $A+Bz+C{z}^2+\cdots$ whose scale is $\alpha ,\beta ,\gamma ,\dots$ (the §63 setup). If the denominator factors into distinct real linear factors $(1-pz)(1-qz)(1-rz)\cdots$, the general term is

$P{z}^n=z^n(U{p}^n+V{q}^n+W{r}^n+\cdots ),$

with $p$ the largest of $|p|,|q|,|r|,\dots$ Then $U{p}^n$ dominates for large $n$, so

$\frac{Q}{P}=\frac{U{p}^{n+1}+V{q}^{n+1}+\cdots }{U{p}^n+V{q}^n+\cdots }⟶p.$

Numerator coefficients $a,b,c,\dots$ only affect $U,V,W,\dots$ — they do not change the limit $p$ (§336). The equation $1-\alpha z-\beta z^2-\cdots =0$ has roots $z=1/p,1/q,\dots$; equivalently $x^m-\alpha x^{m-1}-\beta x^{m-2}-\cdots =0$ (substitute $z=1/x$) has roots $p,q,r,\dots$. So the largest root of the original equation equals the limiting ratio $Q/P$ of the recurrent series whose scale is read off the equation’s coefficients.

Worked Example I (§338): $x^2-3x-1=0$. Form $\frac{1+2z}{1-3z-z^2}$ (numerator $1+2z$ is arbitrary; this happens to give Lucas-like terms). The recurrent series $1,2,7,23,76,251,829,2738,\dots$ gives $2738/829=3.3027744$, agreeing with the true $(3+\sqrt{13})/2=3.3027756$ to six digits.

Example II (§338): $3x-4x^3=1/2$ has three roots $\sin 10°,\sin 50°,\sin 70°$ (since $\sin 3\theta =1/2$ at $\theta =10°,50°,70°$). Substituting $1-6x+8x^3=0$, the smallest root comes from the recurrent series $0,0,1,6,36,208,1200,6912,39808,229248$ with ratio $39808/229248=0.1736515$, vs. true $\sin 10°=0.1736482$.

See bernoullis-method-for-roots.

Movement 2 — Failure modes (§339–§342)

Roots of similar magnitude

Example III (§338) seeks the largest root of $1-6x+8x^3=0$. After $x=y/2$ the equation becomes $y^3-3y+1=0$ with roots $2\sin 10°,2\sin 50°,-2\sin 70°$. The series $1,1,1,2,2,5,4,13,7,35,8,98,-11,\dots$ does not converge cleanly: the two largest roots in absolute value, $-2\sin 70°$ and $2\sin 50°$, are too close in size, and the powers of the second do not vanish quickly compared to the first (§339).

Remedy: substitution shift (§340–§341)

Substitute $x=y-1$ in $1-6x+8x^3=0$ to obtain $8y^3-24y^2+18y-1=0$ with roots $1-\sin 70°,1+\sin 50°,1+\sin 10°$. Now the smallest of these new roots ($\approx 0.06$) is much smaller than the others, so it is reliably extracted by the recurrent series.

Example IV (§340) continues with $z=y/2$, giving $z^3-6z^2+9z-1=0$, recurrent series $1,1,4,31,256,2122,17593,145861$. Ratio $17593/145861=0.12061483$, hence $y=0.06030741$ and $\sin 70°=1-y=0.93969258$, matching the true value to all printed digits.

The general principle (§341): if $k$ is approximately known to be near some root, then $x=y+k$ makes the corresponding $y$ root much smaller than the others, and the smallest-root version of Bernoulli’s method finds it cleanly. This bootstrap to any root is what makes the method universal.

Roots of equal magnitude, opposite sign (§342)

If both $p$ and $-p$ are roots, then $U{p}^n+V(-p{)}^n$ oscillates: $Q/P$ never converges. But alternate ratios $R/P,S/Q,T/R,\dots$ converge to $p^2$. Example: $x^3-x^2-5x+5=0$ has roots $\sqrt{5},-\sqrt{5},1$. The series $1,2,3,8,13,38,63,188,313,938,1563$ has $1563/313\approx 938/188\approx 313/63\approx 5=(\sqrt{5}{)}^2$.

Movement 3 — Numerator choice (§343–§345)

The coefficient $U$ in $P=U{p}^n+\cdots$ depends only on the numerator of the rational function. By choosing the numerator equal to the product of all denominator factors except $(1-pz)$, one cancels that factor entirely and the series becomes geometric in some other root. So a careless choice of numerator can secretly elide the largest root.

§345 fixes this: use numerator $=1$. Then the recurrent series is determined by the scale alone, with no risk of accidental cancellation. The (signed) Example shows $y^3-3y+1=0$ recovered correctly with numerator $1$ and scale $0,3,-1$.

Movement 4 — Repeated real factors (§346–§347)

If the denominator has $(1-pz{)}^2$ with $p$ largest, the general term becomes $z^n((n+1)A{p}^n+B{q}^n+\cdots )$. The ratio $Q/P=(n+2)/(n+1)\cdot p\cdot \frac{(n+1)A+B{q}^n/p^n/(n+1)}{(n+1)A+\cdots }$ still tends to $p$, but slowly and always from above.

Example I (§347): $x^3-3x^2+4=0$ has $x=2$ as a double root. The series $1,3,9,23,57,135,313,711,1593$ has ratios always greater than $2$ — converging slowly because $(n+2)/(n+1)\to 1$ only as $1/n$.

For triple roots $(1-pz{)}^3$ the leading coefficient becomes $\left(\genfrac{}{}{0ex}{}{n+2}{2}\right)A{p}^n$, with the same slow $1+O(1/n)$ excess. The lesson: equations with repeated roots are intrinsically harder for Bernoulli’s method than those with all distinct roots. See bernoullis-method-for-roots for full details.

Movement 5 — Dominant complex factor (§348–§352)

When the denominator has a trinomial factor $1-2pz\cos \varphi +p^2{z}^2$ with $p$ greater than every other denominator pole, the general term (from Chapter 13) is

$P=\frac{A\sin (n+1)\varphi +B\sin n\varphi }{\sin \varphi }{p}^n.$

$Q/P$ involves $\sin (n+2)\varphi /\sin (n+1)\varphi$, which oscillates and never converges. Yet two consecutive recurrent-series relations $P{p}^2+R=2Qp\cos \varphi$ and $Q{p}^2+S=2Rp\cos \varphi$ eliminate the unknown amplitudes $A,B$ and the index $n$ entirely, yielding the closed-form pair

$p=\sqrt{\frac{R^2-QS}{Q^2-PR}},\cos \varphi =\frac{QR-PS}{2\sqrt{(Q^2-PR)(R^2-QS)}}.$

This recovers both modulus $p$ and argument $\varphi$ of the dominant complex conjugate pair from four consecutive terms $P,Q,R,S$ of the recurrent series. See trinomial-factor-from-recurrent-series.

§349 warns of the threshold: the procedure works only when the product of the complex conjugate pair (which equals $p^2$ in the trinomial) is less than the square of the largest real root. Equal magnitudes produce a periodic series; complex roots dominating give nothing at all from a $Q/P$ analysis without the §351–§352 trick.

Example I (§349): $x^3-2x-4=0$ factors as $(x-2)(x^2+2x+2)$. The complex-pair product is $2<2^2=4$, so the real root dominates. Series $1,0,2,4,4,16,24,48,112,192,416,832\to$ ratio $2$. Example II: $x^3-4x^2+8x-8=0$ has real root $2$ and complex pair with product $4=2^2$ (equal magnitude); the series $1,2,2,1,0,0,1,2,2,1,0,0,\dots$ is periodic and no root is read off. Example III: $x^3-3x^2+4x-2=0$ has real root $1$ and complex pair with product $2>1$; the series $1,3,5,5,1,-7,-15,-15,1,33,65,65,1,\dots$ shows the real root $1$ recurring as a coincidence — it is not the dominant root and the method cannot recover it directly.

Movement 6 — Multiple equal trinomial factors and substitution rescue (§353)

When several equal trinomial factors stack, the analysis is impractical. If a real root is already known approximately, the §341 substitution $x=k+y$ converts to an equation in $y$ whose smallest root is the offset; the smallest-root version of Bernoulli’s method handles it.

Example (§353): $x^3-3x^2+5x-4=0$ has a root near $1$. Substituting $x=1+y$ gives $1-2y-y^3=0$. With scale $2,0,1$ the recurrent series is $1,2,4,9,20,44,97,214,472,1041,2296,\dots$, ratio $1041/2296=0.453397$, so $x=1.453397$.

Movement 7 — Geometric progression as a root detector (§354)

If the recurrent series eventually settles into a geometric progression $P,Q,R,S,T,\dots$ with common ratio $x=Q/P$, then because the scale of the relation $\alpha ,\beta ,\gamma ,\delta$ governs $T=\alpha S+\beta R+\gamma Q+\delta P$, substituting $R/P=x^2$, $S/P=x^3$, $T/P=x^4$ produces

$x^4=\alpha x^3+\beta x^2+\gamma x+\delta ,$

i.e. $x$ is a root of the characteristic equation $x^m-\alpha x^{m-1}-\beta x^{m-2}-\cdots =0$. This is the theoretical reason Bernoulli’s method works: the tail of the recurrent series, dominated by the largest root, becomes geometric with ratio $p$, and that ratio satisfies the equation.

Movement 8 — Application to transcendental equations (§355)

The method generalizes formally to infinite equations. Euler’s example: from the sine series,

$\frac{1}{2}=z-\frac{z^3}{6}+\frac{z^5}{120}-\frac{z^7}{5040}+\cdots \text{has smallest root }z=\pi /6.$

Rewriting $0=1-2z+z^3/3-z^5/60+z^7/2520-\cdots$, the scale is $2,0,-1/3,0,1/60,0,-1/2520,0,\dots$. Taking the first few terms of the recurrent series gives $1,2,4,23/3,44/3,1681/60,2408/45$, with ratio $\frac{1681\cdot 45}{2408\cdot 60}=\frac{5043}{9632}=0.52356$, agreeing with $\pi /6=0.523598$ to four digits — error $3/100000$. The method works because all roots of $\sin z=1/2$ are real and the smallest (in absolute value) is well-separated from the next. For most transcendental equations this is not the case, so the method is rarely useful in the transcendental setting.

Significance

Chapter 17 turns the recurrent-series machinery into a general-purpose root-finder:

• Numerically, every root of every polynomial equation is reachable by combining Bernoulli’s method with the substitution shift $x=y+k$. Convergence is governed by the ratio of consecutive root magnitudes — fast when one root dominates, slow when several are close.
• Algorithmically, the method is what we now recognize as the power method for the largest eigenvalue of the companion matrix: iterating $v\mapsto Mv$, the dominant eigenvector emerges and the dominant eigenvalue is the Rayleigh quotient $⟨Mv,v⟩/⟨v,v⟩$ — precisely the $Q/P$ ratio when $v$ is encoded as consecutive terms of the recurrent series.
• The §351–§352 closed form $p=\sqrt{(R^2-QS)/(Q^2-PR)}$, $\cos \varphi =(QR-PS)/(2\sqrt{(Q^2-PR)(R^2-QS)})$ is a 1748 ancestor of techniques for finding complex dominant eigenvalues from real iterations (Wilkinson shifts, Bairstow’s method).

Chapter 17 is the computational counterpart of Chapter 13 — the inverse direction of the same correspondence between the scale of the relation of a recurrent series and the roots of its generating polynomial.

Related pages

• bernoullis-method-for-roots
• trinomial-factor-from-recurrent-series
• recurrent-series
• scale-of-the-relation
• general-term-of-recurrent-series
• closed-form-two-term-recurrence
• trinomial-factor
• real-partial-fraction-decomposition
• sine-and-cosine-series
• chapter-13-on-recurrent-series
• chapter-9-on-trinomial-factors