非线性方程组图形与数值解法

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2016-07-04 18:53

所属专题 Solution of Systems of nonlinear Equations

①本范例演示了如何通过图形和数值方法求解包含两个变量的非线性方程组；②涉及的核心数学概念包括方程组的根、轨迹、隐式曲线以及不同求解方法的比较；③作图要点：使用了 GeoGebra 命令如 Element, Point, If, Locus, Textfield, Polygon, Sequence, RemoveUndefined, Join, Zip, ImplicitCurve 等，对象包括 i, P5, P4, P3, P2, P1, lP, P_i, a, b, Funktion, x_{max}, Px2, x_{min}, Px1, Ax, y_{min}, Py1, y_{max}, Py2, Ay, f00, y0, f1, f2, ε, x0, E0x, g1, g2, E0y, r1, r2, r3, r4, Z1, Z2, N, m1, dx；④适合高中及以上数学学习者，用于理解非线性方程组的解法及比较不同数学工具的应用效果。

GeoGebra 命令及参数

Element(lP, i) → P_i
Point(Segment[Px2, Px1]) → Ax
Point(Segment[Py2, Py1]) → Ay
If(Funktion ≟ 1, a(x, y), If[Funktion ≟ 2, b(x, y), (a(x, y) * b(x, y))]) → f00
If((f1(x0 - ε / 2) * f1(x0 + ε / 2)) < 1E-8 ∧ (f2(x0 - ε / 2) * f2(x0 + ε / 2)) < 1E-8, (x0, y0)) → E0x
If((g1(y0 - ε / 2) * g1(y0 + ε / 2)) < 0 ∧ (g2(y0 - ε / 2) * g2(y0 + ε / 2)) < 0, (x0, y0)) → E0y
If((fxx(xi - ε / 2) * fxx(xi + ε / 2)) < 0, (xi, y(Ay))) → Bxx1
If((fyy(yi - ε / 2) * fyy(yi + ε / 2)) < 0, (x(Ax), yi)) → Byy1
Locus(Bxx1, Ay) → locxx1
Locus(Byy1, Ax) → locyy1
Textfield(a) → inputBox1
Textfield(b) → inputBox2
Polygon((x_{max}, y_{max}), (x_{min}, y_{max}), (x_{min}, y_{min}), (x_{max}, y_{min})) → VV, d, e, g, h
Textfield(N_P) → Textfeld1
If(a_1, N_P, 2) → N_1
Sequence(x_{min} + ((x_{max} - x_{min}) / N_1 * i), i, 0, N_1) → xx, {-6, 0, 6}
Sequence(y_{min} + ((y_{max} - y_{min}) / N_1 * i), i, 0, N_1) → yy, {-6, 0, 6}
RemoveUndefined(Join[Zip[Mirror[{Roots[f00(a, x), y_{min}, y_{max}]}, y = x] + (a, 0), a, xx]]) → Pxxy
Join(Zip[Reverse[{Roots[f00(x, b), x_{min}, x_{max}]}] + (0, b), b, yy]) → Pyyx
Textfield(Funktion) → Textfeld2
Textfield(x_{min}) → Textfeld3
Textfield(y_{min}) → Textfeld4
Textfield(x_{max}) → Textfeld5
Textfield(y_{max}) → Textfeld6
Textfield(N) → Textfeld7
Zip((a(x(pp), y(pp)), b(x(pp), y(pp))), pp, lP) → Liste1
Textfield(ε) → Textfeld8
ImplicitCurve(a) → fa
ImplicitCurve(b) → fb
Textfield(v_{V1}) → InputBox1