In[1]:=

Off[General :: spell1, General :: spell]

In[2]:=

If[$SystemID == "Windows", << A:\MATH375.M, Needs["math375`"]]

Gaussian Elimination

This notebook contains three different algorithms for performing Gaussian Elimination. The first is Gaussian Elimination with nonzero column pivoting. The second is Gaussian Elimination with partial pivoting. The third is Gaussian Elimination with scaled column pivoting.

In[3]:=

a = {{2.0, -1.5, 3.0, 1.0}, {-1.0, 0.0, 2.0, 3.0}, {4.0, -4.5, 5.0, 1.0}} ;

In[4]:=

b = {{1.19, 2.11, -100.0, 1.0, 1.12}, {14.2, -0.122, 12.2, -1.0, 3.44}, {0.0, 100.0, -99.9, 1.0, 2.15}, {15.3, 0.110, -13.1, -1.0, 4.16}} ;

In[5]:=

c = N[{{Pi, -E, Sqrt[2], -Sqrt[3], Sqrt[11]}, {Pi^2, E, -E^2, 3/7, 0}, {Sqrt[5], -Sqrt[6], 1, -Sqrt[2], Pi}, {Pi^3, E^2, -Sqrt[7], 1/9, Sqrt[2]}}] ;

In[6]:=

MatrixForm[a]

Out[6]//MatrixForm=

( 2.` -1.5` 3.` 1.` ) -1.` 0.` 2.` 3.` 4.` -4.5` 5.` 1.`

In[7]:=

MatrixForm[b]

Out[7]//MatrixForm=

( 1.19` 2.11` -100.` 1.` 1.12` ) 14.2` -0.122` 12.2` -1.` 3.44` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

In[8]:=

MatrixForm[c]

Out[8]//MatrixForm=

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 9.869604401089358` 2.718281828459045` -7.3890560989306495` 0.42857142857142855` 0.` 2.23606797749979` -2.449489742783178` 1.` -1.4142135623730951` 3.141592653589793` 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951`

•Nonzero Column Pivoting

In[9]:=

MatrixForm[gencp[a, Verbose -> True]]

Multiply row 1 by -0.5` and subtract from row 2 .

( 2.` -1.5` 3.` 1.` ) 0.` -0.75` 3.5` 3.5` 4.` -4.5` 5.` 1.`

Multiply row 1 by 2.` and subtract from row 3 .

( 2.` -1.5` 3.` 1.` ) 0.` -0.75` 3.5` 3.5` 0.` -1.5` -1.` -1.`

Multiply row 2 by 2.` and subtract from row 3 .

( 2.` -1.5` 3.` 1.` ) 0.` -0.75` 3.5` 3.5` 0.` 0.` -8.` -8.`

Out[9]//MatrixForm=

( 2.` -1.5` 3.` 1.` ) 0.` -0.75` 3.5` 3.5` 0.` 0.` -8.` -8.`

In[10]:=

backsub[gencp[a]]

Out[10]=

{-1.`, 0.`, 1.`}

In[11]:=

MatrixForm[gencp[b, Verbose -> True]]

Multiply row 1 by 11.932773109243698` and subtract from row 2 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

Multiply row 1 by 0.` and subtract from row 3 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

Multiply row 1 by 12.857142857142858` and subtract from row 4 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` 100.` -99.9` 1.` 2.15` 0.` -27.018571428571427` 1272.6142857142859` -13.857142857142858` -10.240000000000002`

Multiply row 2 by -3.952545538971103` and subtract from row 3 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` -1.4210854715202004`*^-14 4664.803967625` -50.11737465946662` -37.07785196089439` 0.` -27.018571428571427` 1272.6142857142859` -13.857142857142858` -10.240000000000002`

Multiply row 2 by 1.0679213396937208` and subtract from row 4 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` -1.4210854715202004`*^-14 4664.803967625` -50.11737465946662` -37.07785196089439` 0.` 0.` -14.740659138451292` -0.04595847236440065` 0.3588052019485062`

Multiply row 3 by -0.00315997397548867` and subtract from row 4 .

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` -1.4210854715202004`*^-14 4664.803967625` -50.11737465946662` -37.07785196089439` 0.` 0.` 0.` -0.2043280720081305` 0.24164015468505834`

Out[11]//MatrixForm=

( 1.19` 2.11` -100.` 1.` 1.12` ) 0.` -25.3001512605042` 1205.47731092437` -12.932773109243698` -9.924705882352944` 0.` -1.4210854715202004`*^-14 4664.803967625` -50.11737465946662` -37.07785196089439` 0.` 0.` 0.` -0.2043280720081305` 0.24164015468505834`

In[12]:=

backsub[gencp[b]]

Out[12]=

{0.1768252974993453`, 0.012692690867687867`, -0.020654050137131123`, -1.1826086954681545`}

In[13]:=

MatrixForm[gencp[c, Verbose -> True]]

Multiply row 1 by 3.141592653589793` and subtract from row 2 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 2.23606797749979` -2.449489742783178` 1.` -1.4142135623730951` 3.141592653589793` 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951`

Multiply row 1 by 0.7117625434171772` and subtract from row 3 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` -0.514718554834473` -0.006584242089740844` -0.18140467425009543` 0.7809433572459716` 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951`

Multiply row 1 by 9.869604401089358` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` -0.514718554834473` -0.006584242089740844` -0.18140467425009543` 0.7809433572459716` 0.` 34.217422396491266` -16.60347971034235` 17.205767384403277` -31.319561065280627`

Multiply row 2 by -0.045720183067485494` and subtract from row 3 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 34.217422396491266` -16.60347971034235` 17.205767384403277` -31.319561065280627`

Multiply row 2 by 3.0393829819641116` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 0.` 19.35831444262284` -0.6353180832051919` 0.3492415164464795`

Multiply row 3 by -35.354897115218506` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 0.` -3.552713678800501`*^-15 2.4395470584060748` 11.117022241648135`

Out[13]//MatrixForm=

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 0.` -3.552713678800501`*^-15 2.4395470584060748` 11.117022241648135`

In[14]:=

backsub[gencp[c]]

Out[14]=

{0.7883937911191892`, -3.125413699353999`, 0.16759659891826414`, 4.557002581008483`}

•Partial Pivoting (AKA Maximal Column Pivoting)

In[15]:=

MatrixForm[gemcp[a, Verbose -> True]]

Swap rows 1 and 3 .

( 4.` -4.5` 5.` 1.` ) -1.` 0.` 2.` 3.` 2.` -1.5` 3.` 1.`

Multiply row 1 by -0.25` and subtract from row 2 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 2.` -1.5` 3.` 1.`

Multiply row 1 by 0.5` and subtract from row 3 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.75` 0.5` 0.5`

Multiply row 2 by -0.6666666666666666` and subtract from row 3 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.` 2.6666666666666665` 2.6666666666666665`

Out[15]//MatrixForm=

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.` 2.6666666666666665` 2.6666666666666665`

In[16]:=

backsub[gemcp[a]]

Out[16]=

{-1.`, 0.`, 1.`}

In[17]:=

MatrixForm[gemcp[b, Verbose -> True]]

Swap rows 1 and 4 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) 14.2` -0.122` 12.2` -1.` 3.44` 0.` 100.` -99.9` 1.` 2.15` 1.19` 2.11` -100.` 1.` 1.12`

Multiply row 1 by 0.9281045751633987` and subtract from row 2 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 -0.22409150326797386` 24.35816993464052` -0.07189542483660127` -0.42091503267973884` 0.` 100.` -99.9` 1.` 2.15` 1.19` 2.11` -100.` 1.` 1.12`

Multiply row 1 by 0.` and subtract from row 3 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 -0.22409150326797386` 24.35816993464052` -0.07189542483660127` -0.42091503267973884` 0.` 100.` -99.9` 1.` 2.15` 1.19` 2.11` -100.` 1.` 1.12`

Multiply row 1 by 0.07777777777777778` and subtract from row 4 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 -0.22409150326797386` 24.35816993464052` -0.07189542483660127` -0.42091503267973884` 0.` 100.` -99.9` 1.` 2.15` -2.220446049250313`*^-16 2.1014444444444442` -98.9811111111111` 1.0777777777777777` 0.7964444444444445`

Swap rows 2 and 3 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` -0.22409150326797386` 24.35816993464052` -0.07189542483660127` -0.42091503267973884` -2.220446049250313`*^-16 2.1014444444444442` -98.9811111111111` 1.0777777777777777` 0.7964444444444445`

Multiply row 2 by -0.0022409150326797387` and subtract from row 3 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` 0.` 24.134302522875817` -0.06965450980392153` -0.4160970653594774` -2.220446049250313`*^-16 2.1014444444444442` -98.9811111111111` 1.0777777777777777` 0.7964444444444445`

Multiply row 2 by 0.021014444444444443` and subtract from row 4 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` 0.` 24.134302522875817` -0.06965450980392153` -0.4160970653594774` -2.220446049250313`*^-16 0.` -96.8817681111111` 1.0567633333333333` 0.751263388888889`

Swap rows 3 and 4 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` 0.` -96.8817681111111` 1.0567633333333333` 0.751263388888889` -2.220446049250313`*^-16 0.` 24.134302522875817` -0.06965450980392153` -0.4160970653594774`

Multiply row 3 by -0.24911088013171717` and subtract from row 4 .

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` 0.` -96.8817681111111` 1.0567633333333333` 0.751263388888889` -2.220446049250313`*^-16 0.` 0.` 0.19359673425367235` -0.22894918134262973`

Out[17]//MatrixForm=

( 15.3` 0.11` -13.1` -1.` 4.16` ) -1.7763568394002505`*^-15 100.` -99.9` 1.` 2.15` 0.` 0.` -96.8817681111111` 1.0567633333333333` 0.751263388888889` -2.220446049250313`*^-16 0.` 0.` 0.19359673425367235` -0.22894918134262973`

In[18]:=

backsub[gemcp[b]]

Out[18]=

{0.17682529749934553`, 0.012692690867687546`, -0.02065405013713109`, -1.182608695468151`}

In[19]:=

MatrixForm[gemcp[c, Verbose -> True]]

Swap rows 1 and 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 9.869604401089358` 2.718281828459045` -7.3890560989306495` 0.42857142857142855` 0.` 2.23606797749979` -2.449489742783178` 1.` -1.4142135623730951` 3.141592653589793` 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554`

Multiply row 1 by 0.3183098861837907` and subtract from row 2 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` 0.36627222260278547` -6.546887300235064` 0.39320366343989627` -0.4501581580785531` 2.23606797749979` -2.449489742783178` 1.` -1.4142135623730951` 3.141592653589793` 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554`

Multiply row 1 by 0.07211662337130922` and subtract from row 3 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` 0.36627222260278547` -6.546887300235064` 0.39320366343989627` -0.4501581580785531` 0.` -2.982363518539235` 1.1908026508341927` -1.422226520525463` 3.039604346745535` 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554`

Multiply row 1 by 0.10132118364233779` and subtract from row 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` 0.36627222260278547` -6.546887300235064` 0.39320366343989627` -0.4501581580785531` 0.` -2.982363518539235` 1.1908026508341927` -1.422226520525463` 3.039604346745535` -4.440892098500626`*^-16 -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711`

Swap rows 2 and 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -2.982363518539235` 1.1908026508341927` -1.422226520525463` 3.039604346745535` -4.440892098500626`*^-16 0.36627222260278547` -6.546887300235064` 0.39320366343989627` -0.4501581580785531`

Multiply row 2 by 0.8602269267144295` and subtract from row 3 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -4.440892098500626`*^-16 -0.2563435308726165` 0.07741457929561535` 0.3098161337288574` -4.440892098500626`*^-16 0.36627222260278547` -6.546887300235064` 0.39320366343989627` -0.4501581580785531`

Multiply row 2 by -0.10564682220388168` and subtract from row 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -4.440892098500626`*^-16 -0.2563435308726165` 0.07741457929561535` 0.3098161337288574` -4.440892098500626`*^-16 5.551115123125783`*^-17 -6.3691593186828666` 0.20902863738304975` -0.1149053997205679`

Swap rows 3 and 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -4.440892098500626`*^-16 -6.3691593186828666` 0.20902863738304975` -0.1149053997205679` -4.440892098500626`*^-16 5.551115123125783`*^-17 -0.2563435308726165` 0.07741457929561535` 0.3098161337288574`

Multiply row 3 by 0.040247624222662405` and subtract from row 4 .

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -4.440892098500626`*^-16 -6.3691593186828666` 0.20902863738304975` -0.1149053997205679` -4.440892098500626`*^-16 5.551115123125783`*^-17 0.` 0.0690016732464472` 0.3144408030779656`

Out[19]//MatrixForm=

( 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951` ) 0.` -3.4669497384023336` 1.6822842168334264` -1.7433087168624704` 3.173334998292711` 0.` -4.440892098500626`*^-16 -6.3691593186828666` 0.20902863738304975` -0.1149053997205679` -4.440892098500626`*^-16 5.551115123125783`*^-17 0.` 0.0690016732464472` 0.3144408030779656`

In[20]:=

backsub[gemcp[b]]

Out[20]=

{0.17682529749934553`, 0.012692690867687546`, -0.02065405013713109`, -1.182608695468151`}

•Scaled Column Pivoting

In[21]:=

MatrixForm[gescp[a, Verbose -> True]]

Swap rows 1 and 3 .

( 4.` -4.5` 5.` 1.` ) -1.` 0.` 2.` 3.` 2.` -1.5` 3.` 1.`

Multiply row 1 by -0.25` and subtract from row 2 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 2.` -1.5` 3.` 1.`

Multiply row 1 by 0.5` and subtract from row 3 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.75` 0.5` 0.5`

Multiply row 2 by -0.6666666666666666` and subtract from row 3 .

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.` 2.6666666666666665` 2.6666666666666665`

Out[21]//MatrixForm=

( 4.` -4.5` 5.` 1.` ) 0.` -1.125` 3.25` 3.25` 0.` 0.` 2.6666666666666665` 2.6666666666666665`

In[22]:=

backsub[gescp[a]]

Out[22]=

{-1.`, 0.`, 1.`}

In[23]:=

MatrixForm[gescp[b, Verbose -> True]]

Swap rows 1 and 2 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 1.19` 2.11` -100.` 1.` 1.12` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

Multiply row 1 by 0.08380281690140845` and subtract from row 2 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 2.120223943661972` -101.02239436619718` 1.0838028169014085` 0.8317183098591551` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

Multiply row 1 by 0.` and subtract from row 3 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 2.120223943661972` -101.02239436619718` 1.0838028169014085` 0.8317183098591551` 0.` 100.` -99.9` 1.` 2.15` 15.3` 0.11` -13.1` -1.` 4.16`

Multiply row 1 by 1.0774647887323945` and subtract from row 4 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 2.120223943661972` -101.02239436619718` 1.0838028169014085` 0.8317183098591551` 0.` 100.` -99.9` 1.` 2.15` 0.` 0.24145070422535214` -26.245070422535214` 0.07746478873239449` 0.4535211267605632`

Swap rows 2 and 3 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 2.120223943661972` -101.02239436619718` 1.0838028169014085` 0.8317183098591551` 0.` 0.24145070422535214` -26.245070422535214` 0.07746478873239449` 0.4535211267605632`

Multiply row 2 by 0.02120223943661972` and subtract from row 3 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 0.` -98.90429064647887` 1.0626005774647886` 0.7861334950704226` 0.` 0.24145070422535214` -26.245070422535214` 0.07746478873239449` 0.4535211267605632`

Multiply row 2 by 0.0024145070422535216` and subtract from row 4 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 0.` -98.90429064647887` 1.0626005774647886` 0.7861334950704226` 0.` -2.7755575615628914`*^-17 -26.003861169014087` 0.07505028169014097` 0.44832993661971815`

Swap rows 3 and 4 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 0.` -26.003861169014087` 0.07505028169014097` 0.44832993661971815` 0.` -2.7755575615628914`*^-17 -98.90429064647887` 1.0626005774647886` 0.7861334950704226`

Multiply row 3 by 3.8034463422044467` and subtract from row 4 .

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 0.` -26.003861169014087` 0.07505028169014097` 0.44832993661971815` 0.` -2.7755575615628914`*^-17 0.` 0.7771508580890086` -0.9190653624665959`

Out[23]//MatrixForm=

( 14.2` -0.122` 12.2` -1.` 3.44` ) 0.` 100.` -99.9` 1.` 2.15` 0.` 0.` -26.003861169014087` 0.07505028169014097` 0.44832993661971815` 0.` -2.7755575615628914`*^-17 0.` 0.7771508580890086` -0.9190653624665959`

In[24]:=

backsub[gescp[b]]

Out[24]=

{0.17682529749934559`, 0.012692690867687552`, -0.02065405013713108`, -1.18260869546815`}

In[25]:=

MatrixForm[gescp[c, Verbose -> True]]

Multiply row 1 by 3.141592653589793` and subtract from row 2 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 2.23606797749979` -2.449489742783178` 1.` -1.4142135623730951` 3.141592653589793` 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951`

Multiply row 1 by 0.7117625434171772` and subtract from row 3 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` -0.514718554834473` -0.006584242089740844` -0.18140467425009543` 0.7809433572459716` 31.006276680299816` 7.3890560989306495` -2.6457513110645907` 0.1111111111111111` 1.4142135623730951`

Multiply row 1 by 9.869604401089358` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` -0.514718554834473` -0.006584242089740844` -0.18140467425009543` 0.7809433572459716` 0.` 34.217422396491266` -16.60347971034235` 17.205767384403277` -31.319561065280627`

Multiply row 2 by -0.045720183067485494` and subtract from row 3 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 34.217422396491266` -16.60347971034235` 17.205767384403277` -31.319561065280627`

Multiply row 2 by 3.0393829819641116` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976` 0.` 0.` 19.35831444262284` -0.6353180832051919` 0.3492415164464795`

Swap rows 3 and 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` 19.35831444262284` -0.6353180832051919` 0.3492415164464795` 0.` 0.` -0.5475426609087787` 0.08697140686311577` 0.30456263781818976`

Multiply row 3 by -0.02828462480716852` and subtract from row 4 .

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` 19.35831444262284` -0.6353180832051919` 0.3492415164464795` 0.` 0.` 0.` 0.06900167324644745` 0.314440803077965`

Out[25]//MatrixForm=

( 3.141592653589793` -2.718281828459045` 1.4142135623730951` -1.7320508075688772` 3.3166247903554` ) 0.` 11.25801605113261` -11.831939037089015` 5.869969521274082` -10.419484076094312` 0.` 0.` 19.35831444262284` -0.6353180832051919` 0.3492415164464795` 0.` 0.` 0.` 0.06900167324644745` 0.314440803077965`

In[26]:=

backsub[gescp[c]]

Out[26]=

{0.7883937911191892`, -3.125413699353999`, 0.16759659891826417`, 4.557002581008483`}

Converted by Mathematica  (November 5, 2002)