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In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding
The s-plane poles and zeros of a 5th-order Chebyshev type II lowpass filter to be approximated as a discrete-time filter The z-plane poles and zeros of the discrete-time Chebyshev filter, as mapped into the z-plane using the matched Z-transform method with T = 1/10 second.
An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities.
Because of this property, zero-based indexing potentially reduces off-by-one and fencepost errors. [8] On the other hand, the repeat count n is calculated in advance, making the use of counting from 0 to n − 1 (inclusive) less intuitive. Some authors prefer one-based indexing, as it corresponds more closely to how entities are indexed in ...
Many applications for Chebyshev nodes, such as the design of equally terminated passive Chebyshev filters, cannot use Chebyshev nodes directly, due to the lack of a root at 0. However, the Chebyshev nodes may be modified into a usable form by translating the roots down such that the lowest roots are moved to zero, thereby creating two roots at ...
Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the ...
A Gram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e −iθ(t), where Hardy's function, Z, is real for real t, and θ is the Riemann–Siegel theta function, we see that zeta is real when sin(θ(t)) = 0.