Own Linear

What are all the linear functions that are their own inverse?

example: the inverse of f(x)=x is f(x)=x. what are all the other LINEAR functions that are their own inverse

f(x) = mx + b
f⁻¹(x) = mx+b

then f⁻¹(f(x)) = x = m(mx+b)+b

x = m²x + mb + b
m²x – x + mb + b = 0
(m² – 1)x + (mb + b) = 0

m²-1 = 0
means m = ± 1

and

mb + b = 0
means if m = 1 then
+b + b = 0
2b =0
b = 0
—- we already knew that f(x) = 1x + 0 = x was a solution

if m = -1 then
-b + b = 0
0 = 0
so any value of b will satisfy the equation.
—- so the other solution is f(x) = -x + b is a solution for any value of b

for example f(x) = -x + 3

f⁻¹(f(x)) = – ( -x + 3 ) + 3 = x -3 +3 = x

Linear equation is a mathematical or algebraic equation that has one or more variables, the equal sign and linear expressions. The iterative method is a good approach in solving linear equations. A wide range of techniques used in obtaining more accurate solutions for linear system.

Linear Equation simple and consists of variables X and Y or any letter in the alphabet, along with equal signs and expressions. Each variable can be a constant or the product of a constant.

Considerations on the use of variables:

• Should not contain exponents; x2
• Do not multiply or divide among themselves; 3xy + 4.
• If you are not under a square root sign.

Thus, linear expression is a statement used in the performance of certain functions addition, subtraction, multiplication and division of numbers. These components can generate a mathematical equation as x + 3, 2x + 5, 3x + 5a.

Learn principles is useful in solving equations. One common form is the equation X + 2 = 5

To find the value of x, let x equal to 1. Both parties should be equal to 5 so that it continues to be true. You must have both a right answer. To balance the equation, both parties must use an equal sign. Conditions is added to a side should be also added to the other side. This is similar to multiplying and dividing both sides of the equation.

The iterative method is used to solve a problem to find the exact solution, based on an initial estimate. The basic idea is repeated a number of measures that generate a response final estimates. Contrast direct methods aimed at resolving the problems through a limited sequence of operations.

The iterative method is useful in solving linear equations involving a large number of variables. The iterative method depends on the pre-conditioners, to improve their performance. Pre-conditioners are the transformation matrix which ensures rapid convergence of the additional costs to overcome its construction. Without it, the method may fail to converge.

The two main classes of iterative methods are:

• Stationary Iterative Method
• And the method is not stationary.

The stationary iterative method can perform the same operation current iteration vectors. Solve a linear system using an operator (a function that operates in another function).

The following is a correction equation based on measurement error, repeating the whole process. The method Stationary is easy to apply and analyze its convergence, but may be limited to a class of matrices (tables of math). Works well with sparse matrices (one mostly populated matrix with zeros) that are easy to parallelize.

The stationary iterative method is one of the oldest methods. It's simple to understand but not as effective. Two examples of this method include

• Method of Jacobi
• and the Gauss-Seidel

The called Jacobi method is considered as an algorithm (finite sequence of instructions) which determines the solution in each row and column, with the highest absolute value. Resolves each diagonal element and plugs into an approximate value. The process is repeated, but convergence is still slow. It is called after Carl Gustav Jakob Jacobi, German mathematician.

On the other hand the Gauss-Seidel was named after Carl Friedrich Gauss and Philipp Ludwig von Seidel. This is an enhanced version of Jacobi. If converges Jacobi, Gauss-Seidel converges more quickly. The method can define the diagonal matrices with nonzero values. Thus, convergence continues to ensure that the matrix can be diagonally dominant and definitely positive.

It refers to the recent steady development of our modern mathematics. It is more difficult to understand, but is very effective. No stationary sequence is based on orthogonal vectors that depend primarily on the co-efficient iteration. Therefore, also goes with the calculations participation data changes at each stage of the iteration. Here are some of the types of method used:

• Conjugate Gradient Method
• MINRAD and SYMMLQ
• CG in the normal equations
• Generalized Minimal Residual
• Gradient BiConjugate
• Quasi minimal residual
• conjugate gradient method Plaza
• BiConjugate gradient stabilized
• Chebyshev Iteration

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Article Source: ArticlesBase.com – Iterative Method: Obtaining Accurate Solutions in Solving Linear Equation