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Linear algebra and geometry
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If the matrix of the system is symmetric and positive definite
Apply the first step of gaussian elimination (without pivoting) to the linear system , where and . How are the matrix and vector modified?
Consider the sequence
,
,
converging to
. Compute the relative errors with respect to for . The minimum relative error is, approximately,for
is not so accurate due to numerical cancellation. Indeed, the relative error associated to it is, approximately:Let is consider a machine working with base , , and rounding to even technique.How many machine numbers with absolute value strictly larger than 1 are there in such machine?
and then with
and . Use both the above formula and an equivalent formula that avoids numerical cancellation. Let be the quantity computed with this second algorithm. Evaluate the relative errors for both the assigned pairs of data. Which of the following statement is correct?
Let us consider a machine working with base , , and rounding to even technique.How many strictly positive machine numbers do exist in such machine?
Generate a vector of linearly spaced elements between and . Multiply the even index components by 2 and the odd index components by . What is the value of the element of index ?
Considering , using which of the following commands the function is correctly represented?