On the Superlinear Convergence of the Secant Method

Vianello, Marco; Zanovello, Renato

doi:10.2307/2324244

Cited by 3 publications

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“…also [18]. For n = 1, it is known that (2) holds with γ equal to the golden mean [31]. In Theorem 6, we show that this result extends to arbitrary n provided F has n − 1 affine component functions and B 0 is initialized exactly.…”

Section: Convergence Of the Broyden-like Updatesmentioning

confidence: 56%

Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

Mannel

2021

Numer Algor

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We consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

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Section: Convergence Of the Broyden-like Updatesmentioning

confidence: 56%

Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

Mannel

2021

Numer Algor

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“…Table 2 Example 1 a: Cumulative runs with α = 0 (top) and αn = 10 −3 (bottom) F is not affine induces a reduction of Algorithm BROY to one dimension, so its convergence rate is the same as that of the one-dimensional secant method, i.e., convergence with exact q-and r-order 1+ √ 5 2 ≈ 1.618 , cf. [20,34]. This implies that the error decays faster than 2-step q-quadratically, which can also be seen from C − 2 and C + 2 .…”

Section: Example 1 Bmentioning

confidence: 75%

On the order of convergence of Broyden’s method

Mannel

2021

Calcolo

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We present two theoretical results and two surprising conjectures concerning convergence properties of Broyden’s method for smooth nonlinear systems of equations. First, we show that when Broyden’s method is applied to a nonlinear mapping $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ F : R n → R n with $$n-d$$ n - d affine component functions and the initial matrix $$B_0$$ B 0 is chosen suitably, then the generated sequence $$(u^k,F(u^k),B_k)_{k\ge 1}$$ ( u k , F ( u k ) , B k ) k ≥ 1 can be identified with a lower-dimensional sequence that is also generated by Broyden’s method. This property enables us to prove, second, that for such mixed linear–nonlinear systems of equations a proper choice of $$B_0$$ B 0 ensures 2d-step q-quadratic convergence, which improves upon the previously known 2n steps. Numerical experiments of high precision confirm the faster convergence and show that it is not available if $$B_0$$ B 0 deviates from the correct choice. In addition, the experiments suggest two surprising possibilities: It seems that Broyden’s method is $$(2d-1)$$ ( 2 d - 1 ) -step q-quadratically convergent for $$d>1$$ d > 1 and that it admits a q-order of convergence of $$2^{1/(2d)}$$ 2 1 / ( 2 d ) . These conjectures are new even for $$d=n$$ d = n .

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