ecuaciones algebraicas lineales. an equation of the form ax+by+c=0 or equivalently ax+by=- c is...
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![Page 1: Ecuaciones Algebraicas lineales. An equation of the form ax+by+c=0 or equivalently ax+by=- c is called a linear equation in x and y variables. ax+by+cz=d](https://reader038.vdocuments.mx/reader038/viewer/2022110207/56649d2d5503460f94a04010/html5/thumbnails/1.jpg)
Ecuaciones Algebraicas lineales
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• An equation of the form ax+by+c=0 or equivalently ax+by=-c is called a linear equation in x and y variables.
• ax+by+cz=d is a linear equation in three variables, x, y, and z.
• Thus, a linear equation in n variables is
a1x1+a2x2+ … +anxn = b
• A solution of such an equation consists of real numbers c1, c2, c3, … , cn. If you need to work more than one linear equations, a system of linear equations must be solved simultaneously.
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Matricesaij = elementos de una matriz
i=número del renglónj=número de la columna
Vector renglón
Vector columna
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Matriz cuadrada m=n
Diagonal principal
Número de incóngnitas
Número deecuaciones
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Reglas de operaciones con matrices
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Representación de ecuaciones algebraicas lineales en forma matricial
Solving for X
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Part 3 10
Noncomputer Methods for Solving Systems of Equations
• For small number of equations (n ≤ 3) linear equations can be solved readily by simple techniques such as “method of elimination.”
• Linear algebra provides the tools to solve such systems of linear equations.
• Nowadays, easy access to computers makes the solution of large sets of linear algebraic equations possible and practical.
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Part 3 11
Gauss EliminationChapter 9
Solving Small Numbers of Equations• There are many ways to solve a system of
linear equations:– Graphical method– Cramer’s rule–Method of elimination– Computer methods
For n ≤ 3
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Part 3 12
Graphical Method
• For two equations:
• Solve both equations for x2:
2222121
1212111
bxaxa
bxaxa
22
21
22
212
1212
11
12
112 intercept(slope)
a
bx
a
ax
xxa
bx
a
ax
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Part 3 13
• Plot x2 vs. x1 on rectilinear paper, the intersection of the lines present the solution.
Fig. 9.1
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Part 3 14
Graphical Method
• Or equate and solve for x1
12
11
22
21
12
1
22
2
12
11
22
21
22
2
12
1
1
22
2
12
11
12
11
22
21
22
21
22
21
12
11
12
112
0
aa
aa
ab
ab
aa
aa
ab
ab
x
a
b
a
bx
a
a
a
a
a
bx
a
a
a
bx
a
ax
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Part 3 15
Figure 9.2
No solution Infinite solutions Ill-conditioned(Slopes are too close)
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Part 3 16
Determinants and Cramer’s Rule
• Determinant can be illustrated for a set of three equations:
• Where A is the coefficient matrix:
bAx
333231
232221
131211
aaa
aaa
aaa
A
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Part 3 17
• Assuming all matrices are square matrices, there is a number associated with each square matrix A called the determinant, D, of A. (D=det (A)). If [A] is order 1, then [A] has one element:
A=[a11]
D=a11
• For a square matrix of order 2, A=
the determinant is D= a11 a22-a21 a12
a11 a12
a21 a22
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Part 3 18
• For a square matrix of order 3, the minor of an element aij is the determinant of the matrix of order 2 by deleting row i and column j of A.
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Part 3 19
223132213231
222113
233133213331
232112
233233223332
232211
333231
232221
131211
aaaaaa
aaD
aaaaaa
aaD
aaaaaa
aaD
aaa
aaa
aaa
D
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Part 3 20
3231
222113
3331
232112
3332
232211 aa
aaa
aa
aaa
aa
aaaD
• Cramer’s rule expresses the solution of a systems of linear equations in terms of ratios of determinants of the array of coefficients of the equations. For example, x1 would be computed as:
D
aab
aab
aab
x 33323
23222
13121
1
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Part 3 23
Method of Elimination
• The basic strategy is to successively solve one of the equations of the set for one of the unknowns and to eliminate that variable from the remaining equations by substitution.
• The elimination of unknowns can be extended to systems with more than two or three equations; however, the method becomes extremely tedious to solve by hand.
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Relación con Cramer
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Part 3 25
Naive Gauss Elimination
• Extension of method of elimination to large sets of equations by developing a systematic scheme or algorithm to eliminate unknowns and to back substitute.
• As in the case of the solution of two equations, the technique for n equations consists of two phases:– Forward elimination of unknowns– Back substitution
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Part 3 26
Fig. 9.3
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Generalizando
Multiplicando ec 1
Restando ec2 de la nueva ec1
Reescribiendo ec anterior
a32’/a22’ = nuevo elemento pivote
Elemento pivote
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Part 3 31
Pitfalls of Elimination Methods
• Division by zero. It is possible that during both elimination and back-substitution phases a division by zero can occur.
• Round-off errors.• Ill-conditioned systems. Systems where small changes
in coefficients result in large changes in the solution. Alternatively, it happens when two or more equations are nearly identical, resulting a wide ranges of answers to approximately satisfy the equations. Since round off errors can induce small changes in the coefficients, these changes can lead to large solution errors.
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Part 3 32
• Singular systems. When two equations are identical, we would loose one degree of freedom and be dealing with the impossible case of n-1 equations for n unknowns. For large sets of equations, it may not be obvious however. The fact that the determinant of a singular system is zero can be used and tested by computer algorithm after the elimination stage. If a zero diagonal element is created, calculation is terminated.
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Part 3 33
Techniques for Improving Solutions
• Use of more significant figures.• Pivoting. If a pivot element is zero,
normalization step leads to division by zero. The same problem may arise, when the pivot element is close to zero. Problem can be avoided:– Partial pivoting. Switching the rows so that the
largest element is the pivot element.– Complete pivoting. Searching for the largest
element in all rows and columns then switching.
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Cramer o sustituciòn
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Determinant Evaluation Using Gauss Elimination
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Casi cero !!!Depende del numero de cifras significativas
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SCALING
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Part 3 44
Gauss-Jordan
• It is a variation of Gauss elimination. The major differences are:–When an unknown is eliminated, it is eliminated
from all other equations rather than just the subsequent ones.
– All rows are normalized by dividing them by their pivot elements.
– Elimination step results in an identity matrix.– Consequently, it is not necessary to employ back
substitution to obtain solution.
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Descomposición LU e inversión de Matrices
[A]{X}={B}
[A]{X}-{B}=0
[U]{X}-{D}=0
Gauss Elimination
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De la eliminación hacia delante de Gauss tenemos :
Finalmente
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Encontrando ‘d’ aplicando la eliminación hacia adelante pero solo sobre el vector ‘B’
Encontrando ‘X’ aplicando la sustitución hacia atrás
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Matriz Inversa
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Homework