1. Select the output display format long and solve the linear system Ax=b, where A is the Hilbert matrix of order n=5,10,15 and b such that the solution x is a vector of all ones. For each n compute the relative error of the solution and the conditioning number using ∝-norm. Comment the results. 2. Write a MATLAB function called elleu which computes L and U factors of the decomposition A=LU. Subsequently, generate the matrix A of order n=100, whose elements are a ij
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=max(i,j) and b such that the solution x is a vector of all ones. Finally, solve the linear system Ax=b, using the decomposition A=LU from the function elleu at first, then by means of the decomposition PA=LU from MATLAB function 1u. In both cases compute the [infinity]-norm of the relative error the solution. Based on the obtained results, deduce what solution is more accurate, motivating your answer. 3. Assemble the matrix A of order n=100, whose elements are a ij
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=imax(i,j). Find the matrices P,L and U from the decomposition PA=LU of the matrix A by means of the MATLAB function 1u. Subsequently, use above factors to invert the matrix A. Verify the result using the MATLAB function inv. 4. Assemble a matrix A of order n=100, whose elements are pseudo-random numbers. Efficiently solve (minimizing the number of arithmetic operations) the following linear systems: ⎩
⎨
⎧
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Ax 1
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=b 1
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Ax 2
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=b 2
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Ax 2
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=b 3
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⋯
Ax 30
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=b 30
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sharing the same matrix A ; let b 1
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such that the corresponding solution x 1
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is a vector of all ones and b i
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=x i−1
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,i=2,…,30. Subsequently, solve each system using MATLAB command \. Comparing the computation time of both procedures, using MATLAB commands tic and toc, and comment the results. 5. Assemble the tridiagonal matrix B of order n=100, whose main diagonal elements are all equal to 10 , while the sub-diagonal and super-diagonal elements are equal to −5 and 5 respectively. Bearing in mind that B is not singular, therefore A=B T
B is symmetric and positive-definite, use the MATLAB function chol to find the Choleski decomposition A=R T
R. After that, use the above decomposition for calculating the inverse of A and for solving the linear system Ax=b, where b such that the solution x is a vector of all ones. Verify the results using MATLAB commands inv and \. 6. Assemble a pseudo-random matrix A of order n, and compute the QR decomposition of A. Later use the factors Q and R for solving the linear system Ax=b, where b such that the solution x is a vector of all ones. Compute the ratio between the computational costs for solving the linear system by means of PA=LU decomposition and QR decomposition, by varying the order of the matrix (for instance n=100,200,…,500 and n=1000,2000,…,5000). Comment the results. 7. Consider the following overdetermined linear system: 1 x 1
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+2x 2
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+3x 3
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+4x 4
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=1
−x 1
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+4x 3
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+x 4
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=2
3x 1
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+5x 2
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+x 3
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=3
2x 1
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−x 2
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+x 4
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=4
x 1
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+x 2
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−x 3
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+x 4
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=5
2x 1
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−x 2
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+3x 4
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=6
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Compute the rank of the matrix of the coefficients of the system. Subsequently, compute the solution of the system in the least-squares sense. Verify the result using the Matlab command \. 8. Implement the Gram-Schmidt orthonormalising method and use it to construct an orthonormal basis of R 5
starting from the following linear independent vectors: v 1
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=(4,2,1,5,−1) T
,v 2
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=(1,5,2,4,0) T
,v 3
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=(3,10,6,2,1) T
v 4
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=(3,1,6,2,−1) T
,v 5
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=(2,−1,2,0,1) T
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Let Q the matrix whose columns are the vectors generated by the procedure. Verify the results of the procedure through Q orthogonality.