<p>Levenberg-Marquardt algorithm</p>

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Решение систем нелинейных уравнений методом Ньютона

Penyelesaian sistem persamaan nonlinear menggunakan metode Newton

in the collapsible area is an implementation of the Levenberg-Marquardt algorithm for nonlinear equation systems.

<p>Levenberg-Marquardt algorithm</p>

Levenberg-Marquardt algorithm x f x.0 ε LM_k.max 100 : k 0 : ν 2 : p x.0 : τ 0.001 : n p f rows : m x.0 rows : ε.1 ε : ε.2 ε : ε.3 ε : ε.p y.0 p f - : j 1 n range k 1 m range J j k el p f j el p k el diff : for for A J transpose J * : I A rows identity : μ τ A Diag max * : stop J transpose p f * normi ε.1 ≤ : y.0 n Zeros : stop ¬ (k k.max < (& k k 1 + : Δp A μ I * + invert J transpose * p f * - : Δp norme ε.2 p norme * ≤ break p.new p Δp + : ρ ε.p norme (2^p f p.new f - norme (2^- (Δp transpose μ Δp * J transpose p f * + (* det / : ρ 0 > p p.new : ε.p p.new f p f - : j 1 n range k 1 m range J j k el p f j el p k el diff : for for A J transpose J * : I A rows identity : stop J transpose p f * normi ε.1 ≤ (: (ε.p norme (2^ε.3 ≤ (| μ μ 13 / 12 ρ * 1 - (3^- Max * : ν 2 : 81 line μ μ ν * : ν 2 ν * : 21 line if 31 line if 21 line 21 line while p k 21 mat 191 line :

mm 1 :

Входные данные:

Data masukan:

Example 2 - input data:

Equations set 2 x f 2 x 3 el (1 -^* x 2 el (1 -^40 * 1 x 2 el 1 x 1 el / x 6 el x 4 el 402^+ sqrt / + (* exp + ln * (x 5 el^* 1.044 mm * - 2 x 3 el (1 -^* x 2 el (1 -^120 * 1 x 2 el 1 x 1 el / x 6 el x 4 el 1202^+ sqrt / + (* exp + ln * (x 5 el^* 2.679 mm * - 2 x 3 el (1 -^* x 2 el (1 -^210 * 1 x 2 el 1 x 1 el / x 6 el x 4 el 2102^+ sqrt / + (* exp + ln * (x 5 el^* 4.749 mm * - 2 x 3 el (1 -^* x 2 el (1 -^20 * 1 x 2 el 1 x 1 el / x 6 el 0.5 - x 4 el 202^+ sqrt / + (* exp + ln * (x 5 el^* 0.18 mm * - 2 x 3 el (1 -^* x 2 el (1 -^120 * 1 x 2 el 1 x 1 el / x 6 el 0.5 - x 4 el 1202^+ sqrt / + (* exp + ln * (x 5 el^* 1.66 mm * - 2 x 3 el (1 -^* x 2 el (1 -^240 * 1 x 2 el 1 x 1 el / x 6 el 0.5 - x 4 el 2402^+ sqrt / + (* exp + ln * (x 5 el^* 4.3 mm * - 61 mat :

Approximation x.0 1007001800475 1.5 0.7 61 mat :

Accuracy ε 1015 -^:

k.max 100 : k 0 : ν 2 : p x.0 : τ 0.001 : n p f rows : m x.0 rows : ε.1 ε : ε.2 ε : ε.3 ε : ε.p y.0 p f - : j 1 n range k 1 m range J j k el p f j el p k el diff : for for A J transpose J * : I A rows identity : μ τ A Diag max * : stop J transpose p f * normi ε.1 ≤ : y.0 n Zeros : stop ¬ (k k.max < (& k k 1 + : Δp A μ I * + invert J transpose * p f * - : Δp norme ε.2 p norme * ≤ break p.new p Δp + : ρ ε.p norme (2^p f p.new f - norme (2^- (Δp transpose μ Δp * J transpose p f * + (* det / : ρ 0 > p p.new : ε.p p.new f p f - : j 1 n range k 1 m range J j k el p f j el p k el diff : for for A J transpose J * : I A rows identity : stop J transpose p f * normi ε.1 ≤ (: (ε.p norme (2^ε.3 ≤ (| μ μ 13 / 12 ρ * 1 - (3^- Max * : ν 2 : 81 line μ μ ν * : ν 2 ν * : 21 line if 31 line if 21 line 21 line while p k μ J A ε.p 61 mat 191 line

Solution sol x f x.0 ε LM_: #

p k μ J A ε.p 61 mat 1007001800475 1.5 0.7 61 mat 7 5.212 108 -^* 6.715 106 -^* - 4.992 1014 -^* - 6.309 107 -^* - 2.487 107 -^* - 0 0.001 0 1.106 1010 -^* - 1.602 106 -^* - 5.303 108 -^* - 0.002 0.002 0 1.104 109 -^* - 2.886 106 -^* - 2.176 108 -^* - 0.005 0.003 1.93 106 -^* - 4.025 1012 -^* - 1.198 107 -^* - 7.456 108 -^* - 0 0.001 0 1.277 109 -^* - 1.019 106 -^* - 1.303 108 -^* - 0.001 0.002 0 5.758 109 -^* - 2.587 106 -^* - 4.606 109 -^* - 0.004 0.003 66 mat 8.942 109 -^* 4.689 1013 -^* 4.083 1010 -^* 5.15 1012 -^* 6.639 107 -^* - 4.535 107 -^* - 4.689 1013 -^* 3.601 1017 -^* 1.956 1014 -^* 7.335 1017 -^* 3.249 1011 -^* - 2.119 1011 -^* - 4.083 1010 -^* 1.956 1014 -^* 1.904 1011 -^* 3.388 1013 -^* 3.047 108 -^* - 2.151 108 -^* - 5.15 1012 -^* 7.335 1017 -^* 3.388 1013 -^* 7.087 1014 -^* 2.285 1010 -^* - 6.326 1010 -^* - 6.639 107 -^* - 3.249 1011 -^* - 3.047 108 -^* - 2.285 1010 -^* - 00 4.535 107 -^* - 2.119 1011 -^* - 2.151 108 -^* - 6.326 1010 -^* - 0066 mat 1.043 2.676 4.744 0.18 1.658 4.295 61 mat 61 mat

A μ I * + invert J transpose * p f * - #