Explain the requirements for calculators. As long as it is a calculator with "Ans" key, it is generally used. The equations to be solved, whether transcendental equations or higher-order equations, are basically the same.

Try it first. If the equation to be solved is exp(x)=-x+3? (Note: exp(x) is the x power of e), the key to be pressed is as follows: 0 =

ln ( - Ans + 3 ) = = =

The ans key has the function of saving the last calculation result, so the first sentence means to assign an initial value to Ans. The initial value should be chosen near the solution and can be roughly estimated. Second, press "=" for more than ten times in a row, and it is found that the value on the screen remains unchanged after pressing it again, which is the solution of the equation. The reason for this is:?

Generally speaking, two-function images have this spiral convergence characteristic near the intersection point.

Assume that the two images in the above figure are y=f(x) and y=g(x) respectively, and the equation to be solved is f(x)=g(x). For convenience, f(x) and g(x) are denoted as the inverse functions of F(x) and G(x) respectively. So this equation can be equivalently transformed into x=F(g(x)) and x=G(f(x)). The right half of these two formulas is to enter the calculator, and then press "=" continuously. Of course, when you enter the calculator, all X's are changed into Ans. Look at the picture above. In fact, of these two formulas, one represents clockwise spiral and the other represents counterclockwise spiral. One can make the spiral converge to the intersection, and the other will make the spiral expand. Unfortunately, we don't know which formula can make the spiral expand and which can make it converge, so we have to try both formulas. If the value on the screen is stable after pressing "=" several times, it means that this is a convergent formula, and this stable value is the solution. For example, in the previous example, the equation can be changed to x=ln(-x+3) and x=-exp(x)+3, where -exp(x)+3 makes the value diffuse and ln(-x+3) makes the value converge, just like at the beginning. ?

If this equation has several solutions, then use different initial values. Generally speaking, it will always converge to a solution closer to the initial value. It should be noted that the spiral direction that makes each solution of the equation converge may be different, that is, for each solution, two formulas are still needed. The above is an ideal situation, such as encountering an equation of x 5+x 2 = x 4-x+5. At this time, it is enough to extract the parts on both sides that best reflect the original characteristics, such as X 5 and X 4. The transformed formula is (X 4-X+5-X 2) under the radical sign of x=5 and (X 5+X 2+X-5) under the radical sign of x=4.