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Linear algebra and geometry
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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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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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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 third component of the solution 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). Let be the matrix obtained by extracting the even columns and odd rows of . The sum of all the elements of is approximately:
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Let be the Hilbert matrix of order 500 (MATLAB command hilb). The number of elements such that is
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Let be the Hilbert matrix of order 50 (MATLAB command hilb). The number of elements such that is
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Let be a vector of linearly spaced values on and a vector of linearly spaced values on . The scalar product between and is, approximately,
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Let be the matrix generated by the command magic(4320), the identity matrix and . Let be the right-hand side such that the solution of the linear system is a vector with all elements equal to 1. Solve the linear system with the MATLAB command \. Let be the infinite norm of the residual . Which is the order of ?
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The maximum value assumed by the function on the interval is, approximately,:
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