Concepts inReal zero isolation for trigonometric polynomials
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complexcoefficients, there is no difference between such a function and a finite Fourier series.
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Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred to using ordinal numbers, as in fourth root, twentieth root, etc. For example: 2 is a square root of 4, since 2 = 4. −2 is also a square root of 4, since (−2) = 4. A real number or complex number has n roots of degree n.
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Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equation where the coefficients ai(x) are polynomial functions of x with rational coefficients. A function which is not algebraic is called a transcendental function.
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Hyperbolic function
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", and so on, corresponding to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh") and so on.
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Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with (sometimes implicit) multiplicity".
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Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in the first three terms respectively have the coefficients 7, −3, and 1.5 (in the third term the variables are hidden, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression).
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Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x − x/4 + 7 is a polynomial, but x − 4/x + 7x is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2).
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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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