<description/> <author>SEL Employee Owner</author> <company/> <keywords/> <revisedBy>SEL Employee Owner</revisedBy> </userData> <identityInfo> <revision>7</revision> <documentID>5ed3af9b-361d-4a70-b31c-13de9017cfcc</documentID> <versionID>652bc72d-3df9-465c-98bd-a37cb8a6deae</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="14" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 2"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="12" font-weight="bold" font-style="italic" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 3"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="12" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Paragraph"> <blockAttr margin-left="0" margin-right="0" text-indent="21" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="List"> <blockAttr margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Indent"> <blockAttr margin-left="108" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="24" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Subtitle" base-style="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="18" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="small" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier New" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="Arial" font-charset="0" font-size="10" font-weight="bold" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="italic" underline="false"/> <mathStyle name="User 5" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="charge" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" imaginary-value="i" zero-threshold="10"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="72" right="72" top="64.8" bottom="64.8"/> <header use-full-page-width="false"> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq2 Arial;}{\f1\fswiss\fprq13 Arial;}{\f2\fnil\fprq10\fcharset0 Arial;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\qc\cf1\f0\fs28\par \f1\fs18\par \cf0\f2\par \par } </center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq13 Arial;}{\f1\fnil\fprq10\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18\par \f1\par \par } </right> </header> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="984" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="977" left="18" top="8.25" width="270" height="12" align-x="38.25" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Triangular Factorization of YBUS discussed in Session 17</u> </b> </p> </text> </region> <region region-id="979" left="264" top="50.25" width="197.25" height="36" align-x="273.75" align-y="60" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that this is different and efficient way compared to the one given the Stevenson and Grainger Book.</p> </text> </region> <region region-id="945" left="24" top="38.25" width="149.25" height="62.25" align-x="54" align-y="72" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">YBUS</ml:id> <ml:matrix rows="4" cols="4"> <ml:imag symbol="i">-4.8</ml:imag> <ml:imag symbol="i">4</ml:imag> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:imag symbol="i">4</ml:imag> <ml:imag symbol="i">-14.5</ml:imag> <ml:imag symbol="i">2.5</ml:imag> <ml:imag symbol="i">8</ml:imag> <ml:real>0</ml:real> <ml:imag symbol="i">2.5</ml:imag> <ml:imag symbol="i">-8.3</ml:imag> <ml:imag symbol="i">5</ml:imag> <ml:real>0</ml:real> <ml:imag symbol="i">8</ml:imag> <ml:imag symbol="i">5</ml:imag> <ml:imag symbol="i">-13</ml:imag> </ml:matrix> </ml:define> </math> <rendering item-idref="1"/> </region> <region region-id="944" left="12" top="132.75" width="92.25" height="13.5" align-x="16.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">LU Factorization</p> </text> </region> <region region-id="732" left="30" top="163.5" width="120" height="87.75" align-x="45.75" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L1</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">YBUS</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">YBUS</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="753" left="30" top="285" width="61.5" height="12.75" align-x="47.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U1</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L1</ml:id> <ml:id xml:space="preserve">YBUS</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="966" left="150" top="302.25" width="150" height="62.25" align-x="166.5" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-8.3</ml:imag> </ml:complex> <ml:complex> <ml:imag>5</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-13</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="4"/> </region> <region region-id="965" left="12" top="410.25" width="132" height="12" align-x="12" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now calculate L2 based on U1</p> </text> </region> <region region-id="762" left="18" top="438.75" width="107.25" height="113.25" align-x="33.75" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L2</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="967" left="36" top="573" width="48.75" height="12.75" align-x="53.25" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U2</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L2</ml:id> <ml:id xml:space="preserve">U1</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="973" left="138" top="572.25" width="165.75" height="62.25" align-x="154.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.7402985074626871</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.26865671641791</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="7"/> </region> <region region-id="969" left="6" top="668.25" width="132" height="12" align-x="6" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now calculate L3 based on U2</p> </text> </region> <region region-id="765" left="24" top="703.5" width="107.25" height="87.75" align-x="39.75" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L3</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U2</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U2</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="970" left="36" top="813" width="48.75" height="12.75" align-x="53.25" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U3</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L3</ml:id> <ml:id xml:space="preserve">U2</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="974" left="138" top="836.25" width="161.25" height="62.25" align-x="154.5" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U3</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> 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fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="72" right="72" top="64.8" bottom="64.8"/> <header use-full-page-width="false"> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq2 Arial;}{\f1\fswiss\fprq13 Arial;}{\f2\fnil\fprq10\fcharset0 Arial;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\qc\cf1\f0\fs28\par \f1\fs18\par \cf0\f2\par \par } </center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq13 Arial;}{\f1\fnil\fprq10\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18\par \f1\par \par } </right> </header> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="984" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="977" left="18" top="8.25" width="270" height="12" align-x="38.25" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <u>Triangular Factorization of YBUS discussed in Session 17</u> </b> </p> </text> </region> <region region-id="979" left="264" top="50.25" width="197.25" height="36" align-x="273.75" align-y="60" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that this is different and efficient way compared to the one given the Stevenson and Grainger Book.</p> </text> </region> <region region-id="945" left="24" top="38.25" width="149.25" height="62.25" align-x="54" align-y="72" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">YBUS</ml:id> <ml:matrix rows="4" cols="4"> <ml:imag symbol="i">-4.8</ml:imag> <ml:imag symbol="i">4</ml:imag> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:imag symbol="i">4</ml:imag> <ml:imag symbol="i">-14.5</ml:imag> <ml:imag symbol="i">2.5</ml:imag> <ml:imag symbol="i">8</ml:imag> <ml:real>0</ml:real> <ml:imag symbol="i">2.5</ml:imag> <ml:imag symbol="i">-8.3</ml:imag> <ml:imag symbol="i">5</ml:imag> <ml:real>0</ml:real> <ml:imag symbol="i">8</ml:imag> <ml:imag symbol="i">5</ml:imag> <ml:imag symbol="i">-13</ml:imag> </ml:matrix> </ml:define> </math> <rendering item-idref="1"/> </region> <region region-id="944" left="12" top="132.75" width="92.25" height="13.5" align-x="16.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">LU Factorization</p> </text> </region> <region region-id="732" left="30" top="163.5" width="120" height="87.75" align-x="45.75" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L1</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">YBUS</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">YBUS</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="753" left="30" top="285" width="61.5" height="12.75" align-x="47.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U1</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L1</ml:id> <ml:id xml:space="preserve">YBUS</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="966" left="150" top="302.25" width="150" height="62.25" align-x="166.5" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-8.3</ml:imag> </ml:complex> <ml:complex> <ml:imag>5</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-13</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="4"/> </region> <region region-id="965" left="12" top="410.25" width="132" height="12" align-x="12" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now calculate L2 based on U1</p> </text> </region> <region region-id="762" left="18" top="438.75" width="107.25" height="113.25" align-x="33.75" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L2</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U1</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="967" left="36" top="573" width="48.75" height="12.75" align-x="53.25" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U2</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L2</ml:id> <ml:id xml:space="preserve">U1</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="973" left="138" top="572.25" width="165.75" height="62.25" align-x="154.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.7402985074626871</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.26865671641791</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="7"/> </region> <region region-id="969" left="6" top="668.25" width="132" height="12" align-x="6" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now calculate L3 based on U2</p> </text> </region> <region region-id="765" left="24" top="703.5" width="107.25" height="87.75" align-x="39.75" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L3</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U2</ml:id> <ml:sequence> <ml:real>3</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U2</ml:id> <ml:sequence> <ml:real>2</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="970" left="36" top="813" width="48.75" height="12.75" align-x="53.25" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U3</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L3</ml:id> <ml:id xml:space="preserve">U2</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="974" left="138" top="836.25" width="161.25" height="62.25" align-x="154.5" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U3</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.7402985074626871</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:complex> <ml:imag>-1.3104512148091017</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="10"/> </region> <region region-id="980" left="18" top="933" width="90" height="18.75" align-x="29.25" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">L1</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">L2</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">L3</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="981" left="162" top="939" width="31.5" height="12.75" align-x="174.75" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <ml:id xml:space="preserve">U3</ml:id> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="982" left="30" top="974.25" width="138.75" height="62.25" align-x="40.5" align-y="1008" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>-0.83333333333333337</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>-0.22388059701492538</ml:real> <ml:real>-0.71641791044776126</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>-0.87736212880833009</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="984" left="222" top="974.25" width="156.75" height="62.25" align-x="234" align-y="1008" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="4"> <ml:complex> <ml:imag>-4.8</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>4</ml:imag> </ml:complex> <ml:complex> <ml:imag>-11.166666666666666</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>2.5</ml:imag> </ml:complex> <ml:complex> <ml:imag>-7.7402985074626871</ml:imag> </ml:complex> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:complex> <ml:imag>8</ml:imag> </ml:complex> <ml:complex> <ml:imag>6.7910447761194028</ml:imag> </ml:complex> <ml:complex> <ml:imag>-1.3104512148091017</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering 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