http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Sun, 14 Jun 2026 00:55:02 GMT Sun, 14 Jun 2026 00:55:02 GMT The latest answers and comments added to the Question, Solving system of linear and nonlinear equations with inequalities http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear?ref=Feed:MaplePrimes:Solving system of linear and nonlinear equations with inequalities:Comments#answer141347 <p>You find the minimum of Ex. In order to that, you don't need to apply the necessary conditions of an extremum. I make use of the penaltymethod=true (see <a href="http://en.wikipedia.org/wiki/Penalty_method">http://en.wikipedia.org/wiki/Penalty_method</a>) option of the Search command. Executing the below a few times (The result of the previous execution is taken as the initialpoint option.), I obtain</p> <form name="worksheet_form"> <table style="width: 576px;" align="center"> <tbody> <tr> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -150;" src="/view.aspx?sf=141347/449801/72efd2e18fc81a67af8cddeb53726b46.gif" alt="restart; interface(rtablesize = 30); PX := [221, 199, 123, 15, 97, 127, 219, 213, 186, 161, 213, 218]; c := [0, 1]; T := [seq(evalf((1/15)*k), k = 0 .. 15)]; m := 7; n := nops(PX)-1; B := proc (n, i, x, t0, t1) options operator, arrow; binomial(n, i)*((x-t0)/(t1-t0))^i*(1-(x-t0)/(t1-t0))^(n-i) end proc; leftX := 1; rightX := 300" width="576" height="167" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/122e4783cc4728d5b4516ef729bad546.gif" alt="EX := sum((sum(PX[k+1]*B(n, k, T[j+1], c[1], c[2]), k = 0 .. n)-(sum(WX[k+1]*B(m, k, T[j+1], c[1], c[2]), k = 0 .. m)))^2, j = 0 .. nops(T)-1):" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/f5bb01f1744a566152d4069530726e8c.gif" alt="var := [seq(WX[k+1], k = 0 .. m), seq(AX[k+1], k = 0 .. m), seq(BX[k+1], k = 0 .. m)]:" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/67be1f541743181453819eef346a0509.gif" alt="cons := {seq(AX[k+1] &gt;= 0, k = 0 .. m), seq(BX[k+1] &gt;= 0, k = 0 .. m), seq(WX[k+1] &gt;= leftX, k = 0 .. m), seq(AX[k+1]*(WX[k+1]-rightX) = 0, k = 0 .. m), seq(BX[k+1]*(leftX-WX[k+1]) = 0, k = 0 .. m), seq(WX[k+1] &lt;= rightX, k = 0 .. m)}:" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -142;" src="/view.aspx?sf=141347/449801/bf739046d9b5fcc4a5fa9c123edff4ce.gif" alt="sol := DirectSearch:-Search(EX, cons, variables = var, evaluationlimit = 200000, penaltymethod = true, initialpoint = [224.730388549617, 165.198497351126, .999904651905370, 74.5125888813888, 300.344632608414, 158.815520182797, 182.894123281704, 219.580143336441, -0.253409086634398e-2, -0.740939531759944e-3, 0.356657080629859e-3, 0.352342061988294e-3, .589982469735124, -0.115051060499348e-3, 0.556001479194536e-3, 0.160881561621582e-2, -0.360031079612710e-3, -0.590949594970267e-3, 223.556348253358, 0.635872033358376e-3, 0.135622291685856e-4, 0.203380375314972e-3, 0.308850965353780e-3, -0.371709295175373e-3])" width="576" height="159" align="middle"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -314;" src="/view.aspx?sf=141347/449801/3c3fbe9b2e76d533b5c97a9da33cd1d9.gif" alt="[100.375231657322, Vector(24, {(1) = 223.31751365457137, (2) = 165.9067899857234, (3) = 1.0000000060044623, (4) = 72.1988574133114, (5) = 300.00000734955927, (6) = 163.7412307542255, (7) = 180.49642781257876, (8) = 219.4484619655704, (9) = 0.539788411621717e-7, (10) = 0.4932695290103134e-7, (11) = -0.3286959523523213e-7, (12) = 0.4411957584880102e-7, (13) = .4594220910156725, (14) = 0.20553636490820796e-6, (15) = -0.9159693528479143e-7, (16) = 0.18619913681988242e-6, (17) = -0.4356533804425423e-7, (18) = 0.1295513797643273e-6, (19) = 324.9018743905167, (20) = 0.13050224312947115e-6, (21) = 0.20103944047224113e-7, (22) = 0.446752828179599e-7, (23) = 0.26662705630527746e-7, (24) = 0.21817410473241325e-7}, datatype = float[8]), 200000]" width="368" height="639"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(1)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/d984ac9eb6a02add5dfab216da7dd867.gif" alt="convert(sol[2], list)" width="126" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -153;" src="/view.aspx?sf=141347/449801/a0ae3eb3dd83517c14c884b9d60dd4a2.gif" alt="[HFloat(223.31751365457137), HFloat(165.9067899857234), HFloat(1.0000000060044623), HFloat(72.1988574133114), HFloat(300.00000734955927), HFloat(163.7412307542255), HFloat(180.49642781257876), HFloat(219.4484619655704), HFloat(5.39788411621717e-8), HFloat(4.932695290103134e-8), HFloat(-3.286959523523213e-8), HFloat(4.411957584880102e-8), HFloat(0.4594220910156725), HFloat(2.0553636490820796e-7), HFloat(-9.159693528479143e-8), HFloat(1.8619913681988242e-7), HFloat(-4.356533804425423e-8), HFloat(1.295513797643273e-7), HFloat(324.9018743905167), HFloat(1.3050224312947115e-7), HFloat(2.0103944047224113e-8), HFloat(4.46752828179599e-8), HFloat(2.6662705630527746e-8), HFloat(2.1817410473241325e-8)]" width="546" height="174" align="middle"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(2)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/972270ced0786abe8e1c35c7001f59ae.gif" alt="``" width="11" height="23"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/cb1be7cde70787a66b5064f5bc3db091.gif" alt="``" width="11" height="23"></p> </td> </tr> </tbody> </table> <input type="hidden" name="sequence" value="1"> <input type="hidden" name="cmd" value="none"></form> <p>It is clear that the values such as -4.35653380442542*10^(-8) should be rounded to 0, 1.00000000600446 should be rounded to 1, and 300.000007349559 should be rounded to 300.</p> <p>PS. This works for Edit. m=11 too.</p> <p>You find the minimum of Ex. In order to that, you don't need to apply the necessary conditions of an extremum. I make use of the penaltymethod=true (see <a href="http://en.wikipedia.org/wiki/Penalty_method">http://en.wikipedia.org/wiki/Penalty_method</a>) option of the Search command. Executing the below a few times (The result of the previous execution is taken as the initialpoint option.), I obtain</p> <form name="worksheet_form"> <table style="width: 576px;" align="center"> <tbody> <tr> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -150;" src="/view.aspx?sf=141347/449801/72efd2e18fc81a67af8cddeb53726b46.gif" alt="restart; interface(rtablesize = 30); PX := [221, 199, 123, 15, 97, 127, 219, 213, 186, 161, 213, 218]; c := [0, 1]; T := [seq(evalf((1/15)*k), k = 0 .. 15)]; m := 7; n := nops(PX)-1; B := proc (n, i, x, t0, t1) options operator, arrow; binomial(n, i)*((x-t0)/(t1-t0))^i*(1-(x-t0)/(t1-t0))^(n-i) end proc; leftX := 1; rightX := 300" width="576" height="167" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/122e4783cc4728d5b4516ef729bad546.gif" alt="EX := sum((sum(PX[k+1]*B(n, k, T[j+1], c[1], c[2]), k = 0 .. n)-(sum(WX[k+1]*B(m, k, T[j+1], c[1], c[2]), k = 0 .. m)))^2, j = 0 .. nops(T)-1):" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/f5bb01f1744a566152d4069530726e8c.gif" alt="var := [seq(WX[k+1], k = 0 .. m), seq(AX[k+1], k = 0 .. m), seq(BX[k+1], k = 0 .. m)]:" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -43;" src="/view.aspx?sf=141347/449801/67be1f541743181453819eef346a0509.gif" alt="cons := {seq(AX[k+1] &gt;= 0, k = 0 .. m), seq(BX[k+1] &gt;= 0, k = 0 .. m), seq(WX[k+1] &gt;= leftX, k = 0 .. m), seq(AX[k+1]*(WX[k+1]-rightX) = 0, k = 0 .. m), seq(BX[k+1]*(leftX-WX[k+1]) = 0, k = 0 .. m), seq(WX[k+1] &lt;= rightX, k = 0 .. m)}:" width="576" height="60" align="middle"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -142;" src="/view.aspx?sf=141347/449801/bf739046d9b5fcc4a5fa9c123edff4ce.gif" alt="sol := DirectSearch:-Search(EX, cons, variables = var, evaluationlimit = 200000, penaltymethod = true, initialpoint = [224.730388549617, 165.198497351126, .999904651905370, 74.5125888813888, 300.344632608414, 158.815520182797, 182.894123281704, 219.580143336441, -0.253409086634398e-2, -0.740939531759944e-3, 0.356657080629859e-3, 0.352342061988294e-3, .589982469735124, -0.115051060499348e-3, 0.556001479194536e-3, 0.160881561621582e-2, -0.360031079612710e-3, -0.590949594970267e-3, 223.556348253358, 0.635872033358376e-3, 0.135622291685856e-4, 0.203380375314972e-3, 0.308850965353780e-3, -0.371709295175373e-3])" width="576" height="159" align="middle"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -314;" src="/view.aspx?sf=141347/449801/3c3fbe9b2e76d533b5c97a9da33cd1d9.gif" alt="[100.375231657322, Vector(24, {(1) = 223.31751365457137, (2) = 165.9067899857234, (3) = 1.0000000060044623, (4) = 72.1988574133114, (5) = 300.00000734955927, (6) = 163.7412307542255, (7) = 180.49642781257876, (8) = 219.4484619655704, (9) = 0.539788411621717e-7, (10) = 0.4932695290103134e-7, (11) = -0.3286959523523213e-7, (12) = 0.4411957584880102e-7, (13) = .4594220910156725, (14) = 0.20553636490820796e-6, (15) = -0.9159693528479143e-7, (16) = 0.18619913681988242e-6, (17) = -0.4356533804425423e-7, (18) = 0.1295513797643273e-6, (19) = 324.9018743905167, (20) = 0.13050224312947115e-6, (21) = 0.20103944047224113e-7, (22) = 0.446752828179599e-7, (23) = 0.26662705630527746e-7, (24) = 0.21817410473241325e-7}, datatype = float[8]), 200000]" width="368" height="639"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(1)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/d984ac9eb6a02add5dfab216da7dd867.gif" alt="convert(sol[2], list)" width="126" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -153;" src="/view.aspx?sf=141347/449801/a0ae3eb3dd83517c14c884b9d60dd4a2.gif" alt="[HFloat(223.31751365457137), HFloat(165.9067899857234), HFloat(1.0000000060044623), HFloat(72.1988574133114), HFloat(300.00000734955927), HFloat(163.7412307542255), HFloat(180.49642781257876), HFloat(219.4484619655704), HFloat(5.39788411621717e-8), HFloat(4.932695290103134e-8), HFloat(-3.286959523523213e-8), HFloat(4.411957584880102e-8), HFloat(0.4594220910156725), HFloat(2.0553636490820796e-7), HFloat(-9.159693528479143e-8), HFloat(1.8619913681988242e-7), HFloat(-4.356533804425423e-8), HFloat(1.295513797643273e-7), HFloat(324.9018743905167), HFloat(1.3050224312947115e-7), HFloat(2.0103944047224113e-8), HFloat(4.46752828179599e-8), HFloat(2.6662705630527746e-8), HFloat(2.1817410473241325e-8)]" width="546" height="174" align="middle"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(2)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/972270ced0786abe8e1c35c7001f59ae.gif" alt="``" width="11" height="23"></p> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=141347/449801/cb1be7cde70787a66b5064f5bc3db091.gif" alt="``" width="11" height="23"></p> </td> </tr> </tbody> </table> <input type="hidden" name="sequence" value="1"> <input type="hidden" name="cmd" value="none"></form> <p>It is clear that the values such as -4.35653380442542*10^(-8) should be rounded to 0, 1.00000000600446 should be rounded to 1, and 300.000007349559 should be rounded to 300.</p> <p>PS. This works for Edit. m=11 too.</p> 141347 Tue, 11 Dec 2012 13:17:07 Z Markiyan Hirnyk Markiyan Hirnyk http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear?ref=Feed:MaplePrimes:Solving system of linear and nonlinear equations with inequalities:Comments#comment141349 <p>Thanks. I've got one question. How do you get your first initialpoint ? Is it something that is random ?</p> <p>Thanks. I've got one question. How do you get your first initialpoint ? Is it something that is random ?</p> 141349 Tue, 11 Dec 2012 17:08:27 Z Przemek Przemek http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear?ref=Feed:MaplePrimes:Solving system of linear and nonlinear equations with inequalities:Comments#comment141350 <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141349">@Przemek</a> when the first time you run.</p> <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141349">@Przemek</a> when the first time you run.</p> 141350 Tue, 11 Dec 2012 17:14:37 Z Markiyan Hirnyk Markiyan Hirnyk http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear?ref=Feed:MaplePrimes:Solving system of linear and nonlinear equations with inequalities:Comments#comment141351 <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141350">@Markiyan Hirnyk</a> To run DirectSearch-Search for the first time I need to pass the parameter initialpoint. How can I get it ?</p> <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141350">@Markiyan Hirnyk</a> To run DirectSearch-Search for the first time I need to pass the parameter initialpoint. How can I get it ?</p> 141351 Tue, 11 Dec 2012 17:26:20 Z Przemek Przemek http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear?ref=Feed:MaplePrimes:Solving system of linear and nonlinear equations with inequalities:Comments#comment141361 <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141351">@Przemek</a> when the first time you run.</p> <p><a href="http://www.mapleprimes.com/questions/141247-Solving-System-Of-Linear-And-Nonlinear#comment141351">@Przemek</a> when the first time you run.</p> 141361 Tue, 11 Dec 2012 20:36:36 Z Markiyan Hirnyk Markiyan Hirnyk