Hessian矩阵的构造

1from manim import *2import numpy as np3 4class BinaryExtremumDemo(ThreeDScene):5    def construct(self):6        # 第一部分：Hessian矩阵的构造7        # 创建标题8        title1 = Text("Hessian矩阵的构造", font="PingFang SC", font_size=36)9        title1.to_edge(UP, buff=0.5)10        self.add_fixed_in_frame_mobjects(title1)11        self.play(Write(title1))12        self.wait()13 14        # 显示一般形式的二元函数15        func = MathTex(16            "z = f(x,y)",17            color=BLUE,18            font_size=3219        )20        func.to_edge(LEFT, buff=1)21        func.shift(UP * 1)22        self.add_fixed_in_frame_mobjects(func)23        self.play(Write(func))24        self.wait()25 26        # 显示Hessian矩阵27        hessian = MathTex(28            "H = \\begin{bmatrix} f_{xx} & f_{xy} \\\\ f_{yx} & f_{yy} \\end{bmatrix}",29            color=YELLOW,30            font_size=3231        )32        hessian.next_to(func, DOWN, buff=1)33        self.add_fixed_in_frame_mobjects(hessian)34        self.play(Write(hessian))35        self.wait()36 37        # 显示判断方法（右侧）38        right_group = VGroup(39            Text("Hessian矩阵的正定性判断：", font="PingFang SC", font_size=28, color=WHITE),40            Text("• 正定矩阵 → 极小值", font="PingFang SC", font_size=28, color=GREEN),41            Text("• 负定矩阵 → 极大值", font="PingFang SC", font_size=28, color=RED),42            Text("• 不定矩阵 → 鞍点", font="PingFang SC", font_size=28, color=YELLOW)43        )44        right_group.arrange(DOWN, aligned_edge=LEFT, buff=0.3)45        right_group.to_edge(RIGHT, buff=1)46        right_group.shift(UP * 1)47        self.add_fixed_in_frame_mobjects(right_group)48        self.play(Write(right_group))49        self.wait(2)50 51        # 淡出第一部分52        self.play(53            *[FadeOut(mob) for mob in self.mobjects],54            run_time=255        )56 57        # 第二部分：3D曲面演示58        # 设置3D场景59        self.set_camera_orientation(phi=75 * DEGREES, theta=30 * DEGREES)60        61        # 创建标题62        title2 = Text("Hessian矩阵判断极值", font="PingFang SC", font_size=36)63        title2.to_corner(UL, buff=0.5)64        self.add_fixed_in_frame_mobjects(title2)65        self.play(Write(title2))66        self.wait()67 68        # 创建3D坐标系69        axes = ThreeDAxes(70            x_range=[-3, 3, 1],71            y_range=[-3, 3, 1],72            z_range=[-2, 10, 2],73            axis_config={"color": GRAY}74        )75        self.add(axes)76        self.wait()77 78        # 创建函数表达式和Hessian矩阵显示79        func = MathTex(80            "f(x,y) = x^2 + 2y^2",81            color=BLUE,82            font_size=3283        )84        func.to_corner(UR, buff=0.5)85        self.add_fixed_in_frame_mobjects(func)86        self.play(Write(func))87        88        hessian = MathTex(89            "H = \\begin{bmatrix} 2 & 0 \\\\ 0 & 4 \\end{bmatrix}",90            color=YELLOW,91            font_size=2892        )93        hessian.next_to(func, DOWN, buff=0.5)94        self.add_fixed_in_frame_mobjects(hessian)95        self.play(Write(hessian))96 97        # 添加判断结果和说明98        judgment_group = VGroup(99            Text("正定矩阵 → 极小值", font="PingFang SC", font_size=28, color=GREEN),100            Text("在点(0,0)处取得极小值", font="PingFang SC", font_size=24, color=WHITE)101        )102        judgment_group.arrange(DOWN, aligned_edge=LEFT, buff=0.3)103        judgment_group.next_to(hessian, DOWN, buff=0.5)104        self.add_fixed_in_frame_mobjects(judgment_group)105        self.play(Write(judgment_group))106        self.wait()107 108        # 创建曲面109        def create_surface(a, c):110            return Surface(111                lambda u, v: np.array([112                    u, v, a*u**2 + c*v**2113                ]),114                u_range=[-3, 3],115                v_range=[-3, 3],116                resolution=(40, 40),117                should_make_jagged=False118            )119 120        # 创建初始曲面 (a=1, c=2, 极小值)121        surface = create_surface(1, 2)122        surface.set_style(fill_opacity=0.8, stroke_width=0.5)123        surface.set_fill_by_checkerboard(ORANGE, BLUE, opacity=0.5)124        self.add(surface)125        self.wait()126 127        # 从极小值(a=1, c=2)到鞍点(a=1, c=-2)128        def update_surface1(surface, alpha):129            c = 2 - 4*alpha130            surface.become(create_surface(1, c))131            surface.set_style(fill_opacity=0.8, stroke_width=0.5)132            surface.set_fill_by_checkerboard(ORANGE, BLUE, opacity=0.5)133 134        def update_hessian1(hessian, alpha):135            c = 2 - 4*alpha136            hessian.become(MathTex(137                f"H = \\begin{{bmatrix}} 2 & 0 \\\\ 0 & {2*c:.1f} \\end{{bmatrix}}",138                color=YELLOW,139                font_size=28140            ))141            hessian.next_to(func, DOWN, buff=0.5)142            self.add_fixed_in_frame_mobjects(hessian)143 144        def update_func1(func, alpha):145            c = 2 - 4*alpha146            func.become(MathTex(147                f"f(x,y) = x^2 + {c:.1f}y^2",148                color=BLUE,149                font_size=32150            ))151            func.to_corner(UR, buff=0.5)152            self.add_fixed_in_frame_mobjects(func)153 154        def update_judgment1(group, alpha):155            if alpha < 0.5:156                group.become(VGroup(157                    Text("正定矩阵 → 极小值", font="PingFang SC", font_size=28, color=GREEN),158                    Text("在点(0,0)处取得极小值", font="PingFang SC", font_size=24, color=WHITE)159                ))160            else:161                group.become(VGroup(162                    Text("不定矩阵 → 鞍点", font="PingFang SC", font_size=28, color=YELLOW),163                    Text("点(0,0)是鞍点", font="PingFang SC", font_size=24, color=WHITE)164                ))165            group.arrange(DOWN, aligned_edge=LEFT, buff=0.3)166            group.next_to(hessian, DOWN, buff=0.5)167            self.add_fixed_in_frame_mobjects(group)168 169        self.play(170            UpdateFromAlphaFunc(surface, update_surface1),171            UpdateFromAlphaFunc(hessian, update_hessian1),172            UpdateFromAlphaFunc(func, update_func1),173            UpdateFromAlphaFunc(judgment_group, update_judgment1),174            run_time=3175        )176        self.wait()177 178        # 从鞍点(a=1, c=-2)到极大值(a=-1, c=-2)179        def update_surface2(surface, alpha):180            a = 1 - 2*alpha181            surface.become(create_surface(a, -2))182            surface.set_style(fill_opacity=0.8, stroke_width=0.5)183            surface.set_fill_by_checkerboard(ORANGE, BLUE, opacity=0.5)184 185        def update_hessian2(hessian, alpha):186            a = 1 - 2*alpha187            hessian.become(MathTex(188                f"H = \\begin{{bmatrix}} {2*a:.1f} & 0 \\\\ 0 & -4 \\end{{bmatrix}}",189                color=YELLOW,190                font_size=28191            ))192            hessian.next_to(func, DOWN, buff=0.5)193            self.add_fixed_in_frame_mobjects(hessian)194 195        def update_func2(func, alpha):196            a = 1 - 2*alpha197            func.become(MathTex(198                f"f(x,y) = {a:.1f}x^2 - 2y^2",199                color=BLUE,200                font_size=32201            ))202            func.to_corner(UR, buff=0.5)203            self.add_fixed_in_frame_mobjects(func)204 205        def update_judgment2(group, alpha):206            if alpha < 0.5:207                group.become(VGroup(208                    Text("不定矩阵 → 鞍点", font="PingFang SC", font_size=28, color=YELLOW),209                    Text("点(0,0)是鞍点", font="PingFang SC", font_size=24, color=WHITE)210                ))211            else:212                group.become(VGroup(213                    Text("负定矩阵 → 极大值", font="PingFang SC", font_size=28, color=RED),214                    Text("在点(0,0)处取得极大值", font="PingFang SC", font_size=24, color=WHITE)215                ))216            group.arrange(DOWN, aligned_edge=LEFT, buff=0.3)217            group.next_to(hessian, DOWN, buff=0.5)218            self.add_fixed_in_frame_mobjects(group)219 220        self.play(221            UpdateFromAlphaFunc(surface, update_surface2),222            UpdateFromAlphaFunc(hessian, update_hessian2),223            UpdateFromAlphaFunc(func, update_func2),224            UpdateFromAlphaFunc(judgment_group, update_judgment2),225            run_time=3226        )227        self.wait()228 229        # 从极大值(a=-1, c=-2)回到极小值(a=1, c=2)230        def update_surface3(surface, alpha):231            a = -1 + 2*alpha232            c = -2 + 4*alpha233            surface.become(create_surface(a, c))234            surface.set_style(fill_opacity=0.8, stroke_width=0.5)235            surface.set_fill_by_checkerboard(ORANGE, BLUE, opacity=0.5)236 237        def update_hessian3(hessian, alpha):238            a = -1 + 2*alpha239            c = -2 + 4*alpha240            hessian.become(MathTex(241                f"H = \\begin{{bmatrix}} {2*a:.1f} & 0 \\\\ 0 & {2*c:.1f} \\end{{bmatrix}}",242                color=YELLOW,243                font_size=28244            ))245            hessian.next_to(func, DOWN, buff=0.5)246            self.add_fixed_in_frame_mobjects(hessian)247 248        def update_func3(func, alpha):249            a = -1 + 2*alpha250            c = -2 + 4*alpha251            func.become(MathTex(252                f"f(x,y) = {a:.1f}x^2 + {c:.1f}y^2",253                color=BLUE,254                font_size=32255            ))256            func.to_corner(UR, buff=0.5)257            self.add_fixed_in_frame_mobjects(func)258 259        def update_judgment3(group, alpha):260            if alpha < 0.4:261                group.become(VGroup(262                    Text("负定矩阵 → 极大值", font="PingFang SC", font_size=28, color=RED),263                    Text("在点(0,0)处取得极大值", font="PingFang SC", font_size=24, color=WHITE)264                ))265            elif alpha < 0.8:266                group.become(VGroup(267                    Text("不定矩阵 → 鞍点", font="PingFang SC", font_size=28, color=YELLOW),268                    Text("点(0,0)是鞍点", font="PingFang SC", font_size=24, color=WHITE)269                ))270            else:271                group.become(VGroup(272                    Text("正定矩阵 → 极小值", font="PingFang SC", font_size=28, color=GREEN),273                    Text("在点(0,0)处取得极小值", font="PingFang SC", font_size=24, color=WHITE)274                ))275            group.arrange(DOWN, aligned_edge=LEFT, buff=0.3)276            group.next_to(hessian, DOWN, buff=0.5)277            self.add_fixed_in_frame_mobjects(group)278 279        self.play(280            UpdateFromAlphaFunc(surface, update_surface3),281            UpdateFromAlphaFunc(hessian, update_hessian3),282            UpdateFromAlphaFunc(func, update_func3),283            UpdateFromAlphaFunc(judgment_group, update_judgment3),284            run_time=3285        )286        self.wait()287 288        # 改变视角以更好地观察289        self.move_camera(phi=60 * DEGREES, theta=60 * DEGREES, run_time=2)290        self.wait()291 292        # 旋转3D视图293        self.begin_ambient_camera_rotation(rate=0.2)294        self.wait(4)295        self.stop_ambient_camera_rotation()296        self.wait()297 298        # 总结299        summary = Text(300            "通过Hessian矩阵的正定性判断极值类型",301            font="PingFang SC",302            color=WHITE,303            font_size=32304        )305        summary.to_edge(DOWN, buff=0.5)306        self.add_fixed_in_frame_mobjects(summary)307        self.play(Write(summary))308        self.wait(2)309 310        # 淡出所有元素311        self.play(312            *[FadeOut(mob) for mob in self.mobjects],313            run_time=2314        )