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Linear algebra and geometry
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Evaluate in .Then, rewrite in order to avoid numerical cancellation and, assuming as exact value the one that is obtained by this last formulation, compute the relative error associated to . Its values is, approximately,
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In a computer the numbers are represented in floating-point arithmetic, with base , digits mantissa and rounding to even technique. Given the numbers and we apply the operation .In which of the following cases we observe numerical cancellation?
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Let a linear system of order , where is a symmetric, tridiagonal matrix with all elements equal to on the main diagonal and to on the superior and inferior codiagonals. The elements of are linearly spaced numbers in . Compute the eigenvalues of and, based on their properties, solve the linear system by the solution of two triangular systems, using the most computationally efficient factorization of . The 1-norm of the vector obtained by summing the vector that is the solution of the inferior triangular system and the one that is the solution of the superior triangular system is, approximately,
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Let a linear system of order , where is a symmetric, tridiagonal matrix with all elements equal to on the main diagonal and to on the superior and inferior codiagonals. The elements of are linearly spaced numbers in . Compute the eigenvalues of and, based on their properties, solve the linear system by the solution of two triangular systems, using the most computationally efficient factorization of . The 1-norm of the vector obtained by summing the vector that is the solution of the inferior triangular system and the one that is the solution of the superior triangular system is, approximately,
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In a computer, the numbers are represented in floating-point arithmetic, with base , digit mantissa and ''rounding to even'' technique. Given the numbers and , we apply the operation . It holds
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Let be the Hilbert matrix of order (MATLAB command hilb) and let be the column vector containing the first strictly positive even numbers. Solve the linear system , using the MATLAB command. The fifth component of the solution is approximately
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The element of the Cholesky factor of the matrix
is:
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Generate the Hilbert matrix of order . Starting from the unit vector, apply iterations of the inverse power method, in order to compute an approximation of the eigenvalue that is the closest to . Then, use the MATLAB command eig to compute the reference ''exact'' value. The relative error associated to the approximation determined by the inverse power method is approximately:
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The maximum value of the function in the interval is approximately
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The maximum value of the function in the interval is approximately
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