Direct methods
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The document discusses the LU factorization method for solving systems of linear equations. It provides an example of applying the Gauss elimination method to a system of 4 equations with 4 unknowns. This results in an upper triangular system that can be easily solved with back substitution. The multipliers used in the row operations are stored in the lower triangular matrix L, while the upper triangular matrix U contains the coefficients from Gauss elimination. The product of L and U yields the original coefficient matrix A, representing the LU factorization of A.
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1. This document discusses methods for solving linear algebraic equations and operations involving matrices. It covers topics such as matrix definitions, types of matrices, matrix operations, representing equations in matrix form, and methods for solving systems of linear equations including graphical methods, determinants, Cramer's rule, elimination, Gauss-Jordan, LU decomposition, and calculating the matrix inverse.
2. Key matrix operations include addition, multiplication, and rules for inverting a matrix. Methods for solving systems of equations include graphical techniques, determinants, Cramer's rule, elimination, Gauss, Gauss-Jordan, and LU decomposition.
3. LU decomposition involves writing a matrix as the product of a lower and upper triangular matrix, which can
Solution of equations for methods iterativos
This document discusses iterative methods for solving systems of equations, including Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. Iterative methods progressively calculate approximations to the solution, unlike direct methods which require completing the full process to obtain the answer. The Jacobi method can solve simple square systems of equations in an iterative fashion. Gauss-Seidel is also an iterative technique that sequentially solves for each unknown using previous approximations. Gauss-Seidel relaxation is similar but incorporates a relaxation parameter. Examples demonstrate applying these methods to solve systems of equations.
Solution of equations for methods iterativos
This document discusses iterative methods for solving systems of equations, including Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. Iterative methods progressively calculate approximations to the solution, unlike direct methods which require completing the full process to obtain the answer. The Jacobi method is used to solve simple square systems of equations. Gauss-Seidel is an iterative technique that solves systems of linear equations by computing updated solutions sequentially using forward substitution. Gauss-Seidel relaxation is similar but incorporates a relaxation parameter. Examples demonstrate applying these methods over multiple iterations to solve systems.
Solution of equations for methods iterativos
This document discusses iterative methods for solving systems of equations. It describes the Jacobi method, which solves systems by iteratively updating solutions. It also describes Gauss-Seidel method, which improves on Jacobi by using previous updated solutions in the current iteration. Both methods are used to progressively calculate better approximations to the solution until reaching an acceptable level of accuracy.
Chapter 3: Linear Systems and Matrices - Part 1/Slides
The document provides information about linear systems and matrices. It begins by defining linear and non-linear equations. It then discusses systems of linear equations, their graphical and geometric interpretations, and the three possible solutions: no solution, a unique solution, or infinitely many solutions. The document also covers matrix notation for representing linear systems, elementary row operations for transforming systems, and determining whether a system has a solution and whether that solution is unique.
Gauss elimination
Gaussian elimination is a method for solving systems of linear equations consisting of two steps:
1) Forward elimination transforms the coefficient matrix into an upper triangular matrix by eliminating variables from lower-numbered equations. This is done by subtracting appropriate multiples of pivot equations from other equations.
2) Back substitution solves the upper triangular system of equations by substituting the solution of higher-numbered equations into lower ones and solving for each variable sequentially, starting from the last equation.
The geometry of three planes in space
This document discusses using systems of linear equations and matrices to represent and find the intersection of planes in three-dimensional space. It provides examples of using the inverse matrix method and reduced row echelon form (RREF) method to solve systems of 2 and 3 planes. The RREF method can find lines of intersection even when planes do not intersect at a single point, and can reveal when planes share a common line of intersection.
Linear Equations
- A linear system includes two or more equations with two or more variables. When two equations are used to model a problem, it is called a linear system.
- Common methods to solve linear systems include graphing the equations to find their intersection point, substitution where one variable is solved for in one equation and substituted into the other, and elimination where equations are combined by multiplication to eliminate a variable.
- The Hill cipher is a method to encrypt plaintext messages by performing matrix multiplication on the message represented as numbers with an encryption key matrix.
Iterativos Methods
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
Iterativos methods
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the solution variables. The Gauss-Seidel method similarly iteratively solves systems but updates the variables in a specific sequential order for increased convergence. Examples are provided of applying both methods through multiple iterations to arrive at solutions. Relaxation is also introduced as a variation of Gauss-Seidel.
System of linear equations 2 eso
This document discusses systems of linear equations and methods for solving them. It defines a system of linear equations as two equations with two unknowns where the solution is a pair of numbers that satisfies both equations. Three methods for solving systems are presented: substitution, where one equation is used to solve for one unknown and substitutes it into the other; matching, where one unknown is solved for in both equations and set equal; and elimination, where the equations are manipulated to eliminate one unknown. The document also defines the three possible types of solutions to a system: a consistent independent system with a single solution, a consistent dependent system with infinite solutions, and an inconsistent system with no solution.
Similar to Direct methods (20)
Some methods for small systems of equations solutions
Some methods for small systems of equations solutions
Chapter 3: Linear Systems and Matrices - Part 1/Slides
Chapter 3: Linear Systems and Matrices - Part 1/Slides
Direct methods
- 1. OSCAR EDUARDO MENDIVELSO OROZCO I study Petroleum Engineering Numerical Methods