I: What is a linear equation
A linear equation, also known as a primary equation, refers to an equation in which the variables are all powers of one, and has the general form ofax+by+…+cz+d=0, the essence of a linear equation is unaffected by multiplying both sides of the equation by any identical nonzero number. Because the representation of any primary equation on the Cartesian coordinate system is a straight line, each term that makes up a primary equation must be a constant or the product of a constant and a variable, and the equation must contain a variable, because if there are no variables and only constants the equation is algebraic and not an equation.
II: What isnonlinear (math.)mathematical equation
A nonlinear equation is a relationship between a dependent variable and an independent variable that is not linear, and there are many such equations, the square relationship x^2, the logarithmic relationship log(x), the exponential relationship 2^x, the trigonometric relationship sin(x), and so on. For example, e^x-con(x)=0 is a nonlinear equation. Compared to linear equations, nonlinear equations are much more complex, both in terms of the existence of solutions and the formulas for solving them.
III: Applications to nonlinear equations
With the rapid development of science and technology, scientific computing is increasingly showing its importance. The application of scientific computing has been widely used in all walks of life, such as the analysis of meteorological data image, aircraft, automobiles and ships shape design, high-tech research and so on can not be separated from scientific computing. Therefore, it is often necessary to require the roots of the nonlinear equation f(x)=0.
For example, we askMissile trajectory yand y is in turn related toThe velocity of the missile v，Spatial location p，Accelerationa，The time parameter tand so on are related, and v can be subdivided intovx,vy,vzThe p can also be divided intopx,py,pzThe variables can not be just a simple linear relationship, such as speed v and time t between the gravity, air resistance, etc., often the final list is a long series of non-linear equations, but the solution of such a system of equations is very meaningful.Often problems in science engineering economics turn on solving systems of nonlinear equations。
IV: Solution of nonlinear equations
1. Real-root pair partitioning
Pairwise division, also known as bisection, sets the function f(x) to be continuous on [a,b] and f(a)f(b)<0, then f(x) has at least one zero on [a,b], which is the mediator theorem in calculus and a prerequisite for the use of pairwise division.
Attention:The pairwise division algorithm is simple, however, if there are several zeros on [a,b], only one of the zeros can be counted; on the other hand, even if there are zeros on [a,b], there may not be f(a)f(b)<0, which restricts the scope of use of the pairwise division method; the pairwise division method can only compute the real roots of the equation.
2. Immovable point iteration method
For a given nonlinear equation f(x)=0, convert it to the equivalent form: x=φ(x). Given an initial value x0, construct the iterative sequence x(k+1)=φ(x(k)),k=1,2,..., if the iterations converge
lim x(k+1)=lim φ(x(k))=α
With α = φ(α),then α is a root of the equation f(x).In the calculation, x(k+1) is taken as the root of the equation when |x(k+1)-x(k)| is less than the given accuracy control quantity.
3. Newton's iterative method
For the nonlinear equation f(x)=0 can be constructed in a variety of iterative formats x(k+1)=φ(x(k)), Newton's iterative method is an iterative format constructed with the help of doing a Taylor expansion of the function f(x) at x0 and taking its linear part.
Do a Taylor expansion of f(x) = 0 at the initial value x0:
f(x)=f(x0)+f′(x0)/(x−x0)+f′′(x0)*(x−x0)^2/2!+…
Taking the linear part of the expansion as an approximation of f(x), the
f(x0)+f′(x0)(x−x0)=0
Assuming that f′(x0) ≠ 0, then
x=x0−f(x0)/f′(x0)
Thus the iterative format of Newton's iterative method is obtained:
x(k+1)=x(k)−f(x(k))/f′(x(k)) , k=1,2,..
4. Chord Intercept Method
In Newton's iterative method: x(k+1)=x(k)-f(x(k))/f′(x(k)),k=1,2,...
The need to solve for the first-order derivative of the function f(x) is usually time-consuming, so the difference quotient can be used instead of the derivative:
f′(x(k))=(f(x(k))−f(x(k-1))) / (x(k)−x(k-1))
Given the initial values x0 and x1, the iterative format of the string-intercept method is:
x(k+1) = x(k)−((x(k)−x(k-1)) * f(x(k)) / (f(x(k))−f(x(k-1)) , k=1,2,…
PS: The above is just a brief introduction to nonlinear equations and will be updated from time to time in the future.