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### Homework Due 9-12-2011 ERO

Course: ISEN 620, Fall 2011
School: Texas A&M
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Word Count: 135

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620 ISEN Homework Due 9-12-2011 Consider the matrix A: 1 3 0 A = 2 1 1 3 9 2 1. Using the following ERO what is the result on A? 2 Row1 + Row2 Row2 3Row1 + Row3 Row3 2. Using the result from 1, continue with EROs, what is the resultant A? 1 / 7 Row2 Row2 1 / 2 Row3 Row3 3. Assume that we have the system of equations Ax = b b, = ( 4 0 10 ) . Apply the T following EROs to solve this system. What is the...

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620 ISEN Homework Due 9-12-2011 Consider the matrix A: 1 3 0 A = 2 1 1 3 9 2 1. Using the following ERO what is the result on A? 2 Row1 + Row2 Row2 3Row1 + Row3 Row3 2. Using the result from 1, continue with EROs, what is the resultant A? 1 / 7 Row2 Row2 1 / 2 Row3 Row3 3. Assume that we have the system of equations Ax = b b, = ( 4 0 10 ) . Apply the T following EROs to solve this system. What is the solution? 2 Row1 + Row2 Row2 3Row1 + Row3 Row3 1 / 7 Row2 Row2 1 / 2 Row3 Row3 4. Using the augmented matrix approach with EROs to obtain the inverse of A: 1 3 0 1 0 0 1 0 0 ? ? ? 2 1 1 0 1 0 0 1 0 ? ? ? 3 9 2 0 0 1 0 0 1 ? ? ?
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Texas A&M - ISEN - 620
Homework: HW Due 9-12-2011Let A be a 3 by 4 matrix: 1 4 2 1A = 3 6 5 1 2 1 3 3Define B as the first three columns of A.The inverse of B is:1 4 2 23 10 32 3 6 5 , B 1 = 1B=711 / 57, B 1 2 1 3 15 9 6 0.404 0.175 0.561 0.018 0.123 0.193
Texas A&M - ISEN - 620
ISEN 620 Homework Due 9-12-2011 Set 25. Redo Problem 4 using the property that matrix multiplication on the left is equivalentto EROs. Develop the matrices that perform the ERO steps you used to solve Problem 4above.
Texas A&M - ISEN - 620
Homework Due 9-19-20111. Solve graphicallymax x0 = 3x1 + 7 x2s.t.2x1 + 4 x2 10-3x1 + 2 x2 3x1 0, x2 02. Solve by the simplex methodmax x0 = 3x1 + 7 x2s.t.2x1 + 4 x2 10-3x1 + 2 x2 3x1 0, x2 0
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Homework Cutting Stock ProblemSolve this cutting stock problem two different ways:(1) Formulate and solve using a SOLVER such as CPLEX or Excel.(2) Solve this formulation via the large scale approach I did in class:max c T B1 P cBs. t. (8, 10, 12) P
Texas A&M - ISEN - 620
Homework for ISEN 620 due Friday 9-2-2011Find the multipliers i ' s such that the given vectors are linear multiples of the basis sets:(a)2 1 5 + 2 3 = 1 1 1 Use the solution to a system of linear equations method to obtain: ( 1 , 2 ) = ( 2, 1)
Texas A&M - ISEN - 620
File: HW 9-12.mleA1.00 4.00 2.00 1.003.00 6.00 -5.00 1.00-2.00 1.00 3.00 3.00B col of A: cfw_ 1, 2, 3B1.00 4.00 2.003.00 6.00 -5.00-2.00 1.00 3.00Binv0.4035 -0.1754 -0.56140.0175 0.1228 0.19300.2632 -0.1579 -0.1053Binv.A1.0000 0.0000 0.0000
Texas A&M - ISEN - 620
ISEN 620 Homework Due 9-12-2011 Set 25. Redo Problem 4 using the property that matrix multiplication on the left is equivalentto EROs. Develop the matrices that perform the ERO steps you used to solve Problem 4above. 1 3 0 1 0 0 2 1 1 0 1 0 3 9 2 0
Texas A&M - ISEN - 620
ISEN 620 Homework Solutions Due 9-12-2011Consider the matrix A:1 3 0A = 2 1 13 9 21. Using the following ERO what is the result on A?2 Row1 + Row2 Row23Row1 + Row3 Row31 3 01 3 0 2 1 1 0 7 1 ERO 3 9 20 0 2 2. Using the result from 1, contin
Texas A&M - ISEN - 620
Homework Due 9-19-2011 solve by the simple methodFile:SimplexAlgEx.mlemax z = 3x1 +7x22x1 + 4x2 &lt;= 10-3x1 + 2x2 &lt;= 3x1 &gt;= 0, x2 &gt;= 0Var z, x1, x2, s1, s2, bdim A: 3 6A1.000 -3.000 -7.000 0.0000.000 2.000 4.000 1.0000.000 -3.000 2.000 0.000rhs:
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Homework Solution for ISEN 620 due Friday 9-2-2011ML Code: 620 Inverse HW.mle1: 2 lam's2 1 5 + 2 3 = 1 1 1 2 1 1 5 1 3 = 1 2 1 2 1 = 1 3 2 1 0.4286 = 2 0.14291 510.1429 5 2 = .2857 1 1 A2.0000 1.0000-1.0000 3.0000b: cfw_
Texas A&M - ISEN - 620
Infeasible RHS PivotsDualSimPiv.mleTv1v2v3v41.000 -2.000 -3.000 -4.0000.000 -1.000 -2.000 -1.0000.000 -2.000 1.000 -3.000v50.0001.0000.000v6rhs0.000 0.0000.000 -3.0001.000 -4.000using row 2 value of -3 to generate pivotTableau - basic
Texas A&M - ISEN - 620
Name:SID:ISEN 620 Test 1 White 10-8-2010Instructions: You may have one 8.5 by 11 inch sheet of notes (front and back) withnormal sized writing or a typed font no less than 10 point. Place this test on top of youranswers and staple the materials toget
Texas A&M - ISEN - 620
Linear Programming Simplex Algorithm Example 1SimpleLP.mle in MORML ModelsData: Ax=b, x&gt;= 0, b&gt;= 0, Maximize(A,b)1.000 -3.000 -6.000 -4.000 0.0000.000 1.000 2.000 1.000 1.0000.000 -2.000 4.000 3.000 0.0000.000 3.000 1.000 2.000 0.000cfw_ x0, x1, x
Texas A&M - ISEN - 620
File: Notes 1 IntroLinSys.DocSolving Linear Systems of EquationsWe will have many situations where we need to solve a system of linear equations.These problems will vary from the smallest two equations and two unknowns toreasonably large systems. Ther
Texas A&M - ISEN - 620
File: Notes 2 Elementary Row Operations 2009.docElementary Row Operations (ERO)An elementary row operation on a matrix A is one of the following operations( Ai represents row i of matrix A):(i)Row i is interchanged with row j: Aj Ai(ii)(iii)Row i
Texas A&M - ISEN - 620
File: Notes 3 GenSolLinSys 2009.DocGeneral Solutions to Linear Systems of EquationsConsider the problem of obtaining a solution to the system of equationsa11 x1 + a12 x2 ++ a1n xn = b1a21 x1 + a22 x2 ++ a2 n x n = b2am1 x1 + am 2 x2 ++ amn xn = bm
Texas A&M - ISEN - 620
File: NonNegLinSys.DocNon-negative Solutions to Linear SystemsConsider a typical constraint set for a linear program. These constraints generally takethe formC y b,y 0.We can convert this system of inequalities into a system of linear equations with
Texas A&M - ISEN - 620
File: EnterVar.docDeciding What Variable Enters the Basis ( x B )The analysis of which variable should enter the basis ( x B ) from the non-basic set ( x N )has been avoided until now. We can select any variable from x N as the entering variableand fi
Texas A&M - ISEN - 620
Simplex Method and the Tableau ProcedureThe solution procedure presented in these notes is the simplex method for solving linearprogramming problems.A key observation about the linear programming solution method (using Basic Feasible Solutions)Tis th
Texas A&M - ISEN - 620
File: One Step Inverse UpdatingConsider that we have an m m matrix B1 that we have computed the inverse (here we areassuming that it exists) B11 . We could have done this inverse by the product form method:B11 = M m M m 1M 2M1Now assume that we want
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Using the LP SOLVER MOR/LPIn the directory list is the mor folder, and within the mor directory is the file mor.exeDouble click on mor.exe and the MOR editor is activated; this is a small editor for writing models andcalling the solver routines.You ca
Texas A&M - ISEN - 620
Approximating Functions of a Single VariableA continuous function f ( x ) can be approximated by a polynomial to any accuracydesired (Weierstrass Approximation Theorem) uniformly over any interval a, b . (Formultiple-dimensional continuous functions th
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
1FINDING SOLUTIONS TO SYSTEMS OF NONLINEAR EQUATIONSFinding a solution to a nonlinear equation such as g( x ) = 0 for x R or to a system ofnonlinear equations such asg1( x ) = 0,g2 ( x ) = 0,Mgn ( x ) = 0,denoted more succinctly as g( x ) = 0 for
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Example N-R Two-dim Root FindingFile: Mult-NR.mleg1[cfw_x1,x2] := 4x1^3+4x1*x2-42x1+2x2^2-14;g2[cfw_x1,x2] := 2x1^2+4x1*x2+4x2^3-26x2 -22;G[cfw_x1,x2] :=cfw_12x1^2+4x2-42, 4x1+4x2cfw_4x1+4x2, 4x1+12x2^2-26;iter: 1xo: cfw_ 4, 3G162.0000 28.000028
Texas A&M - ISEN - 620
Quadratic Forms TestsIn optimization problems it is necessary to test the Hessian matrix, H, for positivedefiniteness, negative-definiteness, or indefinite form. What we are interested in is thefollowing condition (positive-definiteness):y T H y &gt; 0, f
Texas A&M - ISEN - 620
Test 2 Review 620/20101. Cost and rhs sensitivity analysis LP2. Duality LP3. Cutting stock large scale LP4. Definition of local max, min5. Necessary conditions for a local max, min, saddle point (stationary points) FONC6. Sufficient conditions for t
Texas A&M - ISEN - 620
Texas A&M - ISEN - 620
Test 2 Review 620/20101. Cost and rhs sensitivity analysis LP2. Duality LP3. Cutting stock large scale LP4. Definition of local max, min5. Necessary conditions for a local max, min, saddle point (stationary points) FONC6. Sufficient conditions for t
Texas A&M - ISEN - 620
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