Method for estimating numerical adequacy of derivative increments of variables in the damped least-squares automatic lens design problem

Hiroshi Matsui; Kazuo Tanaka

doi:10.1364/AO.34.000642

Applied Optics
Vol. 34,
Issue 4,
pp. 642-647
(1995)
•https://doi.org/10.1364/AO.34.000642

Method for estimating numerical adequacy of derivative increments of variables in the damped least-squares automatic lens design problem

Hiroshi Matsui and Kazuo Tanaka

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Hiroshi Matsui and Kazuo Tanaka, "Method for estimating numerical adequacy of derivative increments of variables in the damped least-squares automatic lens design problem," Appl. Opt. 34, 642-647 (1995)

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Abstract

It is shown that an analytical approach for setting an adequate initial value of a damping factor in the damped least-squares optimization process can also be used to estimate whether the given increments of variables are adequate together with some numerical experiments. It is clarified that the value of a damping factor should almost equal the median of a series of eigenvalues of a squared Jacobian matrix, which is evaluated by use of given increments.

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Case	1/r₁	1/r₂	ϕ₀
1	−1.8	0.5	2126.1
2	−1.9	0.1	3404.6
3	−1.4	−0.7	153.48
4	−0.4	0.5	500.74

Case	δ(1/r₁)	δ(1/r₂)	e_max	e_min
Case 1
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.2 × 10⁻⁴	8.6 × 10⁻⁹
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	1.9 × 10²	9.1 × 10⁻³
Increment 3	1.0 × 10⁰	1.0 × 10⁰	5.2 × 10³	1.5 × 10⁰
Case 2
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.4 × 10⁻⁴	1.4 × 10⁻⁸
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	2.1 × 10²	1.5 × 10⁻²
Increment 3	1.0 × 10⁰	1.0 × 10⁰	5.4 × 10³	2.4 × 10⁰
Case 3
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	1.0 × 10⁻⁵	3.0 × 10⁻¹⁰
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	7.2 × 10⁰	1.9 × 10⁻⁴
Increment 3	2.0 × 10⁰	2.0 × 10⁰	2.1 × 10⁴	1.9 × 10⁰
Case 4
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.0 × 10⁻⁵	2.3 × 10⁻⁷
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	2.1 × 10¹	2.5 × 10⁻¹
Increment 3	1.0 × 10⁰	1.0 × 10⁰	3.2 × 10³	4.8 × 10¹

Variable	Increment 1	Increment 2	Increment 3
δ(1/r₂)	1.0 × 10⁻⁸	1.0 × 10⁻⁴	1.0 × 10⁻⁴
δ(d)	1.0 × 10⁻³	1.0 × 10⁻³	1.0 × 10⁻³
δ(k₁)	1.0 × 10⁻⁷	1.0 × 10⁻⁶	1.0 × 10⁻⁴
δ(A₁′)	6.5 × 10⁻⁸	1.0 × 10⁻⁵	1.0 × 10⁻⁴
δ(B₁)	3.1 × 10⁻⁶	1.0 × 10⁻⁶	1.0 × 10⁻⁴
δ(B₁′)	1.3 × 10⁻⁸	1.0 × 10⁻⁵	1.0 × 10⁻⁶
δ(C₁)	5.5 × 10⁻⁹	1.0 × 10⁻⁵	1.0 × 10⁻⁵
δ(C₁′)	2.1 × 10⁻⁹	1.0 × 10⁻⁶	1.0 × 10⁻⁵
δ(D₁)	8.0 × 10⁻¹⁰	1.0 × 10⁻⁶	1.0 × 10⁻⁶
δ(D₁′)	2.9 × 10⁻¹⁰	1.0 × 10⁻⁷	1.0 × 10⁻⁶
δ(E₁)	1.1 × 10⁻¹⁰	1.0 × 10⁻⁷	1.0 × 10⁻⁶
δ(E₁′)	3.8 × 10⁻¹¹	1.0 × 10⁻⁸	1.0 × 10⁻⁶
δ(F₁)	1.4 × 10⁻¹¹	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(F₁′)	4.8 × 10⁻¹²	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(G₁)	1.7 × 10⁻¹²	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(G₁′)	5.9 × 10⁻¹³	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(H₁)	2.0 × 10⁻¹³	1.0 × 10⁻¹⁰	1.0 × 10⁻⁸
δ(k₂)	1.0 × 10⁻⁸	1.0 × 10⁻⁴	1.0 × 10⁻⁴
δ(A₂′)	2.1 × 10⁻⁹	1.0 × 10⁻⁵	1.0 × 10⁻⁵
δ(B₂)	3.3 × 10⁻¹⁰	1.0 × 10⁻⁴	1.0 × 10⁻⁴
δ(B₂′)	4.7 × 10⁻¹¹	1.0 × 10⁻⁴	1.0 × 10⁻⁵
δ(C₂)	6.1 × 10⁻¹²	1.0 × 10⁻⁴	1.0 × 10⁻⁴
δ(C₂′)	7.8 × 10⁻¹³	1.0 × 10⁻⁴	1.0 × 10⁻⁵
δ(D₂)	9.5 × 10⁻¹⁴	1.0 × 10⁻⁶	1.0 × 10⁻⁵
δ(D₂′)	1.1 × 10⁻¹⁴	1.0 × 10⁻⁶	1.0 × 10⁻⁶
δ(E₂)	1.4 × 10⁻¹⁵	1.0 × 10⁻⁶	1.0 × 10⁻⁶
δ(E₂′)	1.6 × 10⁻¹⁶	1.0 × 10⁻⁸	1.0 × 10⁻⁶
δ(F₂)	1.8 × 10⁻¹⁷	1.0 × 10⁻⁷	1.0 × 10⁻⁶
δ(F₂′)	2.1 × 10⁻¹⁸	1.0 × 10⁻⁹	1.0 × 10⁻⁵
δ(G₂)	2.4 × 10⁻¹⁹	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(G₂′)	2.7 × 10⁻²⁰	1.0 × 10⁻⁹	1.0 × 10⁻⁶
δ(H₂)	3.1 × 10⁻²¹	1.0 × 10⁻⁹	1.0 × 10⁻⁶

Case	1/r₁	1/r₂	ϕ₀
1	−1.8	0.5	2126.1
2	−1.9	0.1	3404.6
3	−1.4	−0.7	153.48
4	−0.4	0.5	500.74

Case	δ(1/r₁)	δ(1/r₂)	e_max	e_min
Case 1
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.2 × 10⁻⁴	8.6 × 10⁻⁹
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	1.9 × 10²	9.1 × 10⁻³
Increment 3	1.0 × 10⁰	1.0 × 10⁰	5.2 × 10³	1.5 × 10⁰
Case 2
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.4 × 10⁻⁴	1.4 × 10⁻⁸
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	2.1 × 10²	1.5 × 10⁻²
Increment 3	1.0 × 10⁰	1.0 × 10⁰	5.4 × 10³	2.4 × 10⁰
Case 3
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	1.0 × 10⁻⁵	3.0 × 10⁻¹⁰
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	7.2 × 10⁰	1.9 × 10⁻⁴
Increment 3	2.0 × 10⁰	2.0 × 10⁰	2.1 × 10⁴	1.9 × 10⁰
Case 4
Increment 1	1.0 × 10⁻⁴	1.0 × 10⁻⁴	2.0 × 10⁻⁵	2.3 × 10⁻⁷
Increment 2	1.0 × 10⁻¹	1.0 × 10⁻¹	2.1 × 10¹	2.5 × 10⁻¹
Increment 3	1.0 × 10⁰	1.0 × 10⁰	3.2 × 10³	4.8 × 10¹

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Applied Optics