Problem 46053. Construct finite difference approximations of derivatives

Created by ChrisR

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f' = (f_{j+1} - f_j})/h\"/></w:customXmlPr><w:r><w:t>f\\prime=\\frac{f_{j+1}-f_j}{h}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\"><w:r><w:t>use Taylor series</w:t></w:r></w:hyperlink><w:r><w:t>. In the example above, one can write</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\"/></w:customXmlPr><w:r><w:t> f_{j+1} = f_j + h f\\prime +  \\frac{h^2}{2}f\\prime\\prime +  \\frac{h^3}{6}f\\prime\\prime\\prime + \\ldots</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Then solving for f' and neglecting terms of order h^2 and higher gives</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f' = (f_{j+1} - f_j)/h - hf''/2\"/></w:customXmlPr><w:r><w:t>f\\prime = \\frac{f_{j+1} - f_j}{h} - \\frac{h}{2}f\\prime\\prime</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Because the exponent on </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"h\"/></w:customXmlPr><w:r><w:t>h</w:t></w:r></w:customXml><w:r><w:t> in the last term is 1, the method is called a first order method.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that takes the order</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> of the derivative and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>terms</w:t></w:r><w:r><w:t> indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>n = 1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>terms = [1 0]</w:t></w:r><w:r><w:t>, and</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ coeffs   = [1 -1]\n errOrder = 1\n errCoeff = -0.5;]]></w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 435.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 217.783px; transform-origin: 407px 217.783px; vertical-align: baseline; "><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.183px 7.91667px; transform-origin: 380.183px 7.91667px; unicode-bidi: normal; "><span style="">In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is</span></span></div><div style="block-size: 37.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.8px; text-align: left; transform-origin: 384px 18.8px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.91667px; transform-origin: 13.6px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f' = (f_{j+1} - f_j})/h" style="width: 84.5px; height: 37.5px;" width="84.5" height="37.5"></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.65px 7.91667px; transform-origin: 365.65px 7.91667px; unicode-bidi: normal; "><span style="">where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf"><span style=""><span style="">use Taylor series</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 116.3px 7.91667px; transform-origin: 116.3px 7.91667px; unicode-bidi: normal; "><span style="">. In the example above, one can write</span></span></div><div style="block-size: 36.0833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.05px; text-align: left; transform-origin: 384px 18.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.91667px; transform-origin: 13.6px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ..." style="width: 226.5px; height: 36px;" width="226.5" height="36"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 214.283px 7.91667px; transform-origin: 214.283px 7.91667px; unicode-bidi: normal; "><span style="">Then solving for f' and neglecting terms of order h^2 and higher gives</span></span></div><div style="block-size: 38.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.3px; text-align: left; transform-origin: 384px 19.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.91667px; transform-origin: 13.6px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f' = (f_{j+1} - f_j)/h - hf''/2" style="width: 130px; height: 38.5px;" width="130" height="38.5"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.3px 7.91667px; transform-origin: 81.3px 7.91667px; unicode-bidi: normal; "><span style="">Because the exponent on </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">h</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.35px 7.91667px; transform-origin: 191.35px 7.91667px; unicode-bidi: normal; "><span style=""> in the last term is 1, the method is called a first order method.</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.717px 7.91667px; transform-origin: 110.717px 7.91667px; unicode-bidi: normal; "><span style="">Write a function that takes the order</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.8px 7.91667px; transform-origin: 91.8px 7.91667px; unicode-bidi: normal; "><span style=""> of the derivative and a vector</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">terms</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.3px 7.91667px; transform-origin: 130.3px 7.91667px; unicode-bidi: normal; "><span style=""> indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">n = 1</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; "><span style=""> and</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">terms = [1 0]</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.91667px; transform-origin: 15.5667px 7.91667px; unicode-bidi: normal; "><span style="">, and</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 7.91667px; transform-origin: 69.3px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> coeffs   = [1 -1]</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> errOrder = 1</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 65.45px 7.91667px; transform-origin: 65.45px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> errCoeff = -0.5;</span></span></div></div></div></div>

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derivative finite difference numerical methods

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Problem 46053. Construct finite difference approximations of derivatives

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Problem 46053. Construct finite difference approximations of derivatives

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