範例：Stiffb 與 Stiffr 的增廣雅可比

函數 > 求解與最佳化 > 微分方程式求解器 > 範例：Stiffb 與 Stiffr 的增廣雅可比

使用 Jacob 函數計算常微分方程式 (ODE) 的增廣雅可比矩陣，然後將其提供作為求解器 Stiffb 與 Stiffr 的輸入。

1. 定義四個未知數系統：

yi 變數是 x 的函數。

2. 定義系統中的常數。

3. 定義初始值。

4. 定義對應至系統右側的向量函數 D(x,y)。

5. 使用 augment 函數建立增廣雅可比：

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6. 呼叫 Stiffb 與 Stiffr 函數：

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傳回的矩陣包含 5 欄，對應至數據點及四個未知數的解。

7. 從傳回的 Stiffb 與 Stiffr 矩陣萃取這四個未知數的解：

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8. 從下列兩個函數繪製並比較每個未知數傳回的解：

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 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yb<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </math>
 <math resultRef="257">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="258">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yr<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </math>
 <math resultRef="259">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="251">
 <real xmlns="http://schemas.mathsoft.com/math50">-0.5</real>
 </startValue>
 <secondTickValue resultRef="253">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </secondTickValue>
 <endValue resultRef="255">
 <real xmlns="http://schemas.mathsoft.com/math50">2.5</real>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

<region id="ID0EWCV5" top="2227.2000000000003" left="38.400000000000006" height="230.40000486373904" width="671.600008368492" xmlns="http://schemas.mathsoft.com/worksheet50">
 <plot background-type="white" origin-positioning="true">
 <xyPlot>
 <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
 <legend />
 <traces>
 <trace resultRef="270">
 <traceStyle color="#FFFF0000" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
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 <trace resultRef="271">
 <traceStyle color="#FF000000" symbol="none" line-weight="1" line-style="DashDotDot">線條</traceStyle>
 </trace>
 </traces>
 <graph-size width="571.600008368492" height="172.800004863739" />
 <axes>
 <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
 <axisLine position="ticknumberlock" positionticmark="0" legendWidth="61.7533333333333" />
 <axisGrid>
 <gridFrequency>5</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
 </numberFormat>
 <plotEquations>
 <plotEquation>
 <math resultRef="274">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">n</id>
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 <math resultRef="275">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
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 <xyDomain scale-type="linear" auto-scale="true">
 <startValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 <secondTickValue resultRef="273">
 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
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 <endValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </endValue>
 </xyDomain>
 </xAxis>
 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="8E-06">
 <axisLine position="ticknumberlock" positionticmark="0" legendWidth="73.69" />
 <axisGrid>
 <gridFrequency>3</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
 </numberFormat>
 <plotEquations>
 <plotEquation>
 <math resultRef="282">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yb<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
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 </math>
 <math resultRef="283">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="284">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yr<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
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 <math resultRef="285">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 <apply xmlns="http://schemas.mathsoft.com/math50">
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 <real>4</real>
 <apply>
 <pow />
 <real>10</real>
 <real>-6</real>
 </apply>
 </apply>
 </secondTickValue>
 <endValue resultRef="281">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <mult />
 <real>8</real>
 <apply>
 <pow />
 <real>10</real>
 <real>-6</real>
 </apply>
 </apply>
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

此繪圖顯示兩個 ODE 求解器傳回相同的解。

相關主題

ODE 求解器

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