PETE 301 Fall 2023 Homework 1 Self study in preparation for Exam 1 Be sure to work textbook example problems in assigned parts of chapters Be sure you can work by hand at least two three steps iterations for provided spreadsheet python examples Note if you are checking answers with MINVERSE it does not accept blanks for zero valued matrix elements recall the A 0 method 1 Problem 9 8a document each row operation during upper triangularization Original augmented matrix 2 6 1 1 2 5 27 61 5 21 5 Augmented matrix after row operations for upper triangularization 2 5 4 1 1 7 0 5 351852 27 53 4 32 11111111 Steps were working top to bottom in each group 10 3 1 10 0 0 0 5 8 6 R1 R1 R2 R2 a21 a11 R1 R3 R3 a31 a11 R1 R1 R1 R2 R2 R3 R3 a32 a22 R2 Answer which can be multiplied by original matrix to obtain original right hand side 2 Problem 9 12 beginning with an augmented matrix Note changed values of b from textbook Steps working from top to bottom or bottom to top in each group as indicated Original augmented matrix Top Down start with R1 R1 R1 a11 R2 R2 a21 R1 R3 R3 a31 R1 2 5 3 1 0 0 1 2 1 0 5 0 5 0 5 1 2 1 0 5 4 5 2 5 1 4 5 0 5 6 5 3 5 Top Down R1 R1 R2 R2 a22 R3 R3 a32 R2 Bottom Up start with R3 R1 R1 a13 R3 a12 R2 R2 R2 a23 R3 R3 R3 a33 3 LU Doolittle decomposition 1 0 0 1 0 0 0 5 1 0 0 1 0 0 5 9 2 0 0 1 0 5 13 10 14 32 5 Steps U11 A11 U12 A12 U13 A13 L21 A21 U11 L31 A31 U11 U22 A22 L21 U12 U23 A23 L21 U13 L32 A32 L31 U12 U22 U33 A33 L31 U13 L32 U23 1 0 375 0 25 0 1 0 4 0 0 1 L U 8 0 0 2 6 25 0 1 1 625 8 1 X where UX D Ax LUx b Define Ux D so LD b 1 forward subst of L with b to get D 2 backward subst of U with D to get x 4 Problem 10 11b using Forward substitution to solve for D where LD B followed by Backward substitution for D L 1b 10 50 666663 10 91093333 x U 1D 0 999956667 4 99969667 3 000476662 5 Problem 11 13 Perform only two iterations with calculator Original A augmented R1 R3 R2 R1 R3 R2 Reordered rows of A for diagonal dominance augmented 2 3 8 8 2 3 6 1 1 1 6 1 1 7 2 2 1 7 x3 38 34 20 20 38 34 Es1 0 7 166667 8 155754 7 99168 0 2 7619 1 94076 1 99919 0 388201 0 020389 Es2 x1 Gauss Seidel to Es 5 0 Use initial x1 x2 x3 0 Iter x2 0 1 2 3 0 2 5 4 08631 4 00466 Es3 Es max 0 121275 0 020531 0 423105 0 029228 0 423105 0 029228 Over relaxation to Es 5 0 Use initial x1 x2 x3 0 lambda iter 1 2 over relaxation parameter is sub optimal x3 x2 x3 x1 x2 x1 Es1 0 1 2 3 4 5 6 2 5 4 294 3 908 4 034 3 987 4 005 0 3 4 553 3 779 4 085 3 968 4 012 7 333 8 314 7 847 8 07 7 97 8 012 0 8 8 8 217 7 773 8 129 7 938 8 027 2 314 1 732 2 127 1 945 2 023 1 991 0 2 777 1 523 2 248 1 885 2 05 1 979 0 341 0 205 0 075 0 029 0 011 Es2 Es3 Es max 0 071 0 057 0 044 0 024 0 011 0 824 0 323 0 193 0 081 0 036 0 824 0 323 0 193 0 081 0 036 Optimal Relaxation with 0 968 Use initial x1 x2 x3 0 lambda 0 968 under relaxation parameter determined using solver to minimize Es max after 3 its Es max iter Es3 Es2 Es1 x2 x1 x3 x2 x1 x3 0 1 2 5 2 4 05 3 4 014 0 2 421 3 998 4 013 7 14 8 123 8 007 0 6 914 8 085 8 009 2 832 1 989 1 993 0 2 742 2 013 1 994 0 145 0 009 0 363 0 009 0 395 0 009 0 395 0 004 6 Problem 17 16 matrix solution larger than 2x2 not required for exam be sure you can set up larger problems such as this 3x3 case 7 Problem 17 17 You may use Excel to calculate needed summations Don t forget to answer the last question a Compute summations needed for Eqs 17 6 and 17 7 using Excel formulas then calculate slope intercept and r2 using Excel formulas You may check your work with the Excel trend line tool on your required plot and or SLOPE and INTERCEPT functions b Use SLOPE INTERCEPT and RSQ functions You may check your work with the Excel trend line tool on c Use SLOPE INTERCEPT and RSQ functions You may check your work with the Excel trend line tool on d Use MINVERSE with Eq 17 19 You may check your work with the Excel trend line tool on your required your required plot your required plot plot 8 Problem 20 56 see 17 1 5 Linearization of Nonlinear Relationships solve by hand using Eqs 17 6 17 7 You may check your work with SLOPE INTERCEPT and RSQ functions This type of polymer water mixture is often used for hydraulic fracturing and or to improve mobility ratio for water injection