23 Sentences With "finding the roots of" | Random Sentence Generator

The study is a first step in what has proved a difficult area of genetic research: finding the roots of complex behavior.

Finding the roots of risk in the brain also "helps us understand what might be making people different in terms of their risk appetites," he said.

This method is useful for finding the roots of polynomials of high degree to arbitrary precision; it has almost optimal complexity in this setting.

For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned. However, the problem of finding the roots of a polynomial can be very ill- conditioned. Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not. For the problem of solving the linear equation where is invertible, the condition number is given by , where is the operator norm subordinate to the normal Euclidean norm on .

Sir Leonard Bairstow, CBE, FRS, FRAeS (25 June 1880 – 8 September 1963) was an English aeronautical engineer. Bairstow is best remembered for his work in aviation and for Bairstow's method for arbitrarily finding the roots of polynomials.

A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method is an iterative method for finding the roots of a differentiable function , which are solutions to the equation .

Consequently, several parallel graph algorithms utilizing pointer jumping have been designed. These include algorithms for finding the roots of a forest of rooted trees, connected components, minimum spanning trees, and biconnected components. However, pointer jumping has also shown to be useful in a variety of other problems including computer vision, image compression, and Bayesian inference.

In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow. The algorithm finds the roots in complex conjugate pairs using only real arithmetic. See root-finding algorithm for other algorithms.

He worked in many areas of computational and applied mathematics, ranging from software development for nuclear reactors modeling to approximation by polynomials, from quadrature on a sphere to numerical solution of stiff for which he developed explicit Chebyshev methods called DUMKA, systems of PDEs, from domain decomposition and Poincaré–Steklov operators to Finite difference methods, from iterative solvers to parallel computing. He even contributed to finding the roots of a cubic equation.

Once the interpolating polynomial p_{n,k} (x) has been calculated, one can also calculate the next approximation x_{n+k+1} as a solution of p_{n,k} (x)=0 instead of using (). For k = 1 these two methods are identical: it is the secant method. For k = 2 this method is known as Muller's method. For k = 3 this approach involves finding the roots of a cubic function, which is unattractively complicated.

Dr. Susheela P. Upadhyaya, an eminent scholar has made a comprehensive study in finding the roots of Beary literature. Dr. A. Wahhab Doddamane has produced a book entitled The Muslims of Dakshina Kannada, which is an informative documentary work. The Bearys have also produced a number of magazines and periodicals from Mangalore and other cities of the district. Some periodicals have become popular and a few of them have become a part of Beary history.

Johnson-Davies also contributed several ground-breaking articles to the BBC Micro magazine, Acorn User. In May 1986, he explored the infinite graphical potential of Benoit Mandelbrot’s mathematics in "Join the Mandelbrot Set". In July 1986, in "Back to the Roots", Johnson-Davies applied the Newton–Raphson method for finding the roots of an equation to create some stunning images that displayed fractal behaviour. In the October 1986 issue, he wrote "Spider Power" with quantum computing pioneer David Deutsch.

Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent. The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:. # "Guess" a zero a of the polynomial f.

The results of a factor analysis can be used to estimate each individual's score on the primary abilities based upon the individual's scores on the tests. Chapter X presents a method for obtaining the regression weights for estimating primary abilities from subject scores, and well as for estimating subjects scores from the primary traits (for estimating the components of variance of the subject scores). Appendices. I: Outline of Calculations for the Centroid Method with Unknown Diagonals. II: A Method of Finding the Roots of a Polynomial.

Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials.

The function \tau is only an abstract representation of a computation which, in practice, may be relatively complex. Some methods result in a \tau which is a closed-form continuous function while others need to be decomposed into a series of computational steps involving, for example, SVD or finding the roots of a polynomial. Yet another class of methods results in \tau which must rely on iterative estimation of some parameters. This means that both the computation time and the complexity of the operations involved may vary between the different methods.

However, this iterative scheme is numerically unstable; the approximation errors accumulate during the successive factorizations, so that the last roots are determined with a polynomial that deviates widely from a factor of the original polynomial. To reduce this error, one may, for each root that is found, restart Newton's method with the original polynomial, and this approximate root as starting value. However, there is no warranty that this will allow finding all roots. In fact, the problem of finding the roots of a polynomial from its coefficients is in general highly ill-conditioned.

A toy program is a small computer program typically used for educational purposes. Toy programs are generally of little practical use, although the concepts implemented may be useful in a much more sophisticated program. A toy program typically focuses on a specific problem, such as computing the Nth term in a sequence, finding the roots of a quadratic equation and testing if a number is prime. Toy programs are also used for a developer trying out a new programming language, to test all of the language's syntax and coding methods.

Finding the roots of a given polynomial has been a prominent mathematical problem. Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted in 1799 and completely proved in 1824. This result also holds for equations of higher degrees.

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth degree polynomial: :x^6 - 9 x^3 + 8 = 0.

Corchado also stated that his dream of a career in journalism was largely based on the hopes of finding the roots of his homeland. According to different accounts, Corchado's parents were supportive of his journalism, but did not want him to report on drug trafficking. Corchado has said that he tried to avoid writing about the drug wars “until the issue was something you couldn't ignore anymore.” Corchado worked on the U.S.-Mexico border for Public Radio, later becoming a reporter for the Standard-Examiner in Ogden, Utah; the El Paso Herald-Post; and The Wall Street Journal, based in its Philadelphia and Dallas bureaus.

The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. This is easy for 2 \times 2 matrices, but the difficulty increases rapidly with the size of the matrix. In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).

In mathematics and computing, a root-finding algorithm is an algorithm for finding zeroes, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeroes of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeroes, expressed either as floating point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function .