Rearrange the equations to form a strictly diagonally dominant system. Use the Jacobi iterative m...
Rearrange the equations to form a strictly diagonally dominant system. Use the Jacobi iterative method and Gauss-Seidel methods with an initial vector (0, 0, 0) and 10 iterations to approximate the solution of the system.
Solve the system directly and compare your results.
X - 8Y - 2Z = 1
X + Y + 5Z = 4
3X - Y +Z = -2
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Rearrange the equations to form a strictly diagonally dominant system. Use the Jacobi iterative m...
Please show all steps clearly. thanks 1. (40 points) Rearrange the equations to form a strictly...
Please show all steps clearly. thanks
1. (40 points) Rearrange the equations to form a strictly diagonally dominant system. Apply two steps of the Jacobi and Gauss-Seidal Methods from starting vector(0,0,0". [i -8 -21 1 1 5 (3 -11
Fundamentals: Jacobi and Gauss-Seidel Methods Consider the 4-equations for 4-unknowns, written in matrix form at right....
Fundamentals: Jacobi and Gauss-Seidel Methods Consider the 4-equations for 4-unknowns, written in matrix form at right. Reorder the equations to form a new Ax b problem where the new matrix A is "strictly diagonally dominant" (or at least the "best you can do" to make as "strong" a diagonal as possible). -5 3 4 2x2 3 3 14 -1-212」(x,
2. 3x 25」LX2 (a) Perform three iterations for the following iterative methods using initial guess x0. Compute relative residual for each iteration. (You can use a calculator) · Jacobi method » Gauss...
2. 3x 25」LX2 (a) Perform three iterations for the following iterative methods using initial guess x0. Compute relative residual for each iteration. (You can use a calculator) · Jacobi method » Gauss-Seidel method · SOR method with ω 1.2 (b) For each iterative method, express its iteration procedure in the following matrix form: In other words, determine B and c for (2).
2. 3x 25」LX2 (a) Perform three iterations for the following iterative methods using initial guess x0. Compute relative...
Test II. ITERATIVE SOLUTION OF SYSTEMS OF LINEAR EQUATIONS Solve the following linear system using Gauss-Seidel...
Test II. ITERATIVE SOLUTION OF SYSTEMS OF LINEAR EQUATIONS Solve the following linear system using Gauss-Seidel iterative method. Use x = x; = x; =0 as initial guesses. Perform two iterations of the method to find xị, xį and xſ and fill the following table. Show all the calculation steps. 10x, + 2x2 - X3 = 27 -3x, - 6x2 + 2xz = -61.5 X1 + x2 + 5x3 = -21.5
Consider the linear system 5x1 - 21 + X1 - 22 + x3 = 1 5.22...
Consider the linear system 5x1 - 21 + X1 - 22 + x3 = 1 5.22 - 23 = 2 22 5 5x3 = 3 (a) Discuss the convergence of the iterative solutions of this system generated by the Jacobi and Gauss-Seidel methods, by considering their iterative matrices. (b) If both methods converge, which one of them converges faster to the exact solution of this system? (c) Starting with the initial approximation x(0) = [0,0,0], find the number of iterations...
just 1,2,4 Problem 1 Consider the linear system of equations Ax = b, where x €...
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
QUESTION 2 Consider the linear system T 0.50 + 0.2 -1.425 12 0.5x2 + 0.25.73 23...
QUESTION 2 Consider the linear system T 0.50 + 0.2 -1.425 12 0.5x2 + 0.25.73 23 whose solution is (0.9,-0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) = (0,0,0)' as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x) = (0,0,0) [20]
QUESTION 2 Consider the linear system Ti 0.521 + 21 2 0.5x2 + 13 0.25.13 23...
QUESTION 2 Consider the linear system Ti 0.521 + 21 2 0.5x2 + 13 0.25.13 23 0.2 -1.425 2 whose solution is (0.9,-0.8, 0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) - (0,0,0)' as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x(0)...
QUESTION 2 Consider the linear system 11 0.50 + Ii 22 0.579 + 23 0.2533 13...
QUESTION 2 Consider the linear system 11 0.50 + Ii 22 0.579 + 23 0.2533 13 0.2 -1.425 2 whose solution is (0.9,-0.8.0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) = (0,0,0) as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w x() = (0,0,0) = 0.7 and...
QUESTION 2 i 0.2 -1.425 2 Consider the linear system 13 0.52 + 22 0.25.13 =...
QUESTION 2 i 0.2 -1.425 2 Consider the linear system 13 0.52 + 22 0.25.13 = Ii 0.522 + 23 whose solution is 0.9,-0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x() = (0,0,0)* as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x(0)...
Please show all steps clearly. thanks
1. (40 points) Rearrange the equations to form a strictly diagonally dominant system. Apply two steps of the Jacobi and Gauss-Seidal Methods from starting vector(0,0,0". [i -8 -21 1 1 5 (3 -11
Fundamentals: Jacobi and Gauss-Seidel Methods Consider the 4-equations for 4-unknowns, written in matrix form at right. Reorder the equations to form a new Ax b problem where the new matrix A is "strictly diagonally dominant" (or at least the "best you can do" to make as "strong" a diagonal as possible). -5 3 4 2x2 3 3 14 -1-212」(x,
2. 3x 25」LX2 (a) Perform three iterations for the following iterative methods using initial guess x0. Compute relative residual for each iteration. (You can use a calculator) · Jacobi method » Gauss-Seidel method · SOR method with ω 1.2 (b) For each iterative method, express its iteration procedure in the following matrix form: In other words, determine B and c for (2).
2. 3x 25」LX2 (a) Perform three iterations for the following iterative methods using initial guess x0. Compute relative...
Test II. ITERATIVE SOLUTION OF SYSTEMS OF LINEAR EQUATIONS Solve the following linear system using Gauss-Seidel iterative method. Use x = x; = x; =0 as initial guesses. Perform two iterations of the method to find xị, xį and xſ and fill the following table. Show all the calculation steps. 10x, + 2x2 - X3 = 27 -3x, - 6x2 + 2xz = -61.5 X1 + x2 + 5x3 = -21.5
Consider the linear system 5x1 - 21 + X1 - 22 + x3 = 1 5.22 - 23 = 2 22 5 5x3 = 3 (a) Discuss the convergence of the iterative solutions of this system generated by the Jacobi and Gauss-Seidel methods, by considering their iterative matrices. (b) If both methods converge, which one of them converges faster to the exact solution of this system? (c) Starting with the initial approximation x(0) = [0,0,0], find the number of iterations...
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
QUESTION 2 Consider the linear system T 0.50 + 0.2 -1.425 12 0.5x2 + 0.25.73 23 whose solution is (0.9,-0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) = (0,0,0)' as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x) = (0,0,0) [20]
QUESTION 2 Consider the linear system Ti 0.521 + 21 2 0.5x2 + 13 0.25.13 23 0.2 -1.425 2 whose solution is (0.9,-0.8, 0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) - (0,0,0)' as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x(0)...
QUESTION 2 Consider the linear system 11 0.50 + Ii 22 0.579 + 23 0.2533 13 0.2 -1.425 2 whose solution is (0.9,-0.8.0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) = (0,0,0) as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w x() = (0,0,0) = 0.7 and...
QUESTION 2 i 0.2 -1.425 2 Consider the linear system 13 0.52 + 22 0.25.13 = Ii 0.522 + 23 whose solution is 0.9,-0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x() = (0,0,0)* as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x(0)...
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