Base shifting, splicing, and deflating are three techniques used in numerical analysis and linear algebra. These techniques are used to manipulate and analyze data and are particularly useful in solving problems related to matrices, vectors, and linear equations.
Base Shifting: Base shifting is a technique used to convert a polynomial from one base to another. This technique is particularly useful in polynomial interpolation, where we want to approximate a function using a polynomial of a given degree.
Suppose we have a polynomial f(x) of degree n, defined as:
We can rewrite this polynomial in terms of a new base (x – c), where c is some constant, as follows:
The coefficients b_i can be calculated using a recursive formula known as the Horner’s method. Once we have the polynomial expressed in terms of the new base, we can evaluate it at any point x using the standard polynomial evaluation algorithm. Base shifting is also used in numerical integration, where we want to approximate the value of a definite integral using a finite number of function evaluations. In this case, we can shift the integration limits to a new range (a, b), where a and b are constants, and then apply a numerical integration method such as the trapezoidal rule or Simpson’s rule.
Splicing: Splicing is a technique used to combine two or more matrices or vectors into a single entity. This technique is useful in solving systems of linear equations, where we want to express the system in matrix form and then apply matrix algebra to find the solution. Suppose we have two matrices A and B, each of size n x m, and we want to splice them into a single matrix C of size n x 2m. We can do this by concatenating the columns of A and B, as follows:
where | denotes the horizontal concatenation operator. The resulting matrix C has the same number of rows as A and B, but twice as many columns. Splicing can also be used to combine two or more vectors into a single vector. Suppose we have two vectors u and v, each of size n, and we want to splice them into a single vector w of size 2n. We can do this by concatenating the elements of u and v, as follows:
where, denotes the vertical concatenation operator. The resulting vector w has twice as many elements as u and v.
Deflating: Deflating is a technique used to reduce the degree of a polynomial by one. This technique is useful in polynomial root-finding algorithms, where we want to find the roots of a polynomial of a given degree.
Suppose we have a polynomial f(x) of degree n, defined as:
and we have found one of its roots, say x = r. We can use the deflation technique to reduce the degree of the polynomial by one, as follows:
where q(x) is a polynomial of degree n-1. We can find the coefficients of q(x) by dividing f(x) by (x – r) using polynomial long division. The resulting polynomial g(x) has one less root than f(x), and we can repeat the deflation process on g(x) to find the remaining roots.
Deflation can also be used in polynomial interpolation, where we want to approximate a function using a polynomial of a given degree. Suppose we have a set of n+1 data points (x_i, y_i), and we want to find a polynomial of degree n that passes through all the points. We can use the Lagrange interpolation formula to find the coefficients of the polynomial, but this can lead to numerical instability for large values of n. To overcome this problem, we can use the deflation technique to find a sequence of polynomials of decreasing degree that approximate the function with increasing accuracy.
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1. What is base shifting in index numbers? |
2. How is splicing used in index numbers? |
3. What does deflating mean in the context of index numbers? |
4. How does base shifting affect the interpretation of index numbers? |
5. Why is splicing important in index numbers analysis? |
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