Linear Algebra II (finish from last time) Root finding

Transcription

1 Lecture 5: Topics: Linear lgebra II (finish from last time) Root finding -Cross product of two vectors -Finding roots of linear equations -Finding roots of nonlinear equations HW: HW 1, Part 3-4 given in class *Remember: ll parts of HW 1 are due on 1/31 or 2/1 Feb 2/3 : TEST 1 in class Team declarations due by

2 b Why solve multiple linear equations? Steady state mass balance analysis Set derivative term (accumulation rate) to 0 Then given concentration data find k i values Given k i values find concentration values Static force eample Sum of forces in and y directions 0 Eqns. for resolution of force into and y components Given some known values, solve for unknowns

3 Equation Classes 1. Multiple variables, linear b (-b)0 2. Scalar (1 var) and nonlinear f() 0 3. Multiple variables, nonlinear 2 +y 2 4, y 2 4. Differential equations dc dt ( t) 0.5C ( t) dc dt ( t) C t C v f (, r) T f (, y) ( t + t) C t ( t) 5. Optimization min f ( ) s. t. g( ) < 0

4 Review N Specify the magnitude and direction of the net force acting on the femur in this hypothetical loading scenario. 45 N 75 N

5 Step 1: Represent each vector in its Cartesian components +y F 1 [75j] N F 2 [-45j] N F 3 [-100cos45i 100sin45j] N + F N Step 2: dd like components F R ([-100cos45]i + [ sin45]j) N F R ([-70.7]i + [-40.7]j) N +y F 2 45 N F R θ + Step 3: Find the magnitude and angle of the resultant vector F R ( ) 1/ N Θ tan -1 (40.7/70.7) N F 1

6 Matri division: Not defined B

7 Matri Inversion: In arithmetic there are three fundamental concepts that are often needed to conduct operations. 1. dditive identity 2. Multiplicative identity 3. Inverse Scalar arithmetic: - Zero is an additive identity. You add zero to any other number and you get that number back. - The number 1 is a multiplicative identity. 1 times any other number gives you the number back. - The reciprocal of a number is it s inverse. number times its inverse is q q 1 q q 1 q 1 q We have the equivalents in matri arithmetic. The zero matri is the additive identity. The identity matri is the multiplicative identity. matri can have an inverse, -1. *Not all matrices have inverses, just as zero doesn t have a reciprocal.

8 Two rules for matri inversion 1. Not all matrices have inverses. (invertible matrices have non-zero determinants) 2. Only square matrices can have inverses. d c b a a c b d 1 The inverse of a 2 2 matri is easily calculated where is the determinant of the matri

9 Eample ( 6) Check ( -1 I) In MTLB Verify direct and indirect matri inversion: Direct: -1 inv()

10 Recap of Linear lgebra lecture material Topics Covered Why we model/simulate Biomedical Systems rrays: Scalars, Vectors, Matrices Basic MTLB operations Cartesian vectors (used to represent physical vector quantities) Cartesian vector operations (2D and 3D) For Test 1 material related to Linear lgebra, be able to: Define basic terms (review lecture notes) Perform vector operations (dot and cross product) Perform matri operations (determinant, matri math, solutions of b) Know basic MTLB synta (Go back and review eamples done in lecture) Resolve Cartesian vectors into components dd Cartesian vectors Develop simple mass balances

11 Why is it important to be able to find the roots of an equation If we represent a system/process by a mathematical equation, then the roots of that equation represent a system/process state that corresponds to zero response : y 3 *This equation can be though to describe the response of variable y to changes in variable The nature of the response can be linear or nonlinear (most biological processes are nonlinear) Eamples of nonlinear processes: Kinetics of enzyme reactions Population migration of cells the Dunn Eq Bioheat transport Flow-mediated transport

12 Finding the roots of a linear equation f() m +b f() : Some function of variable m : The slope of the linear function b : The value of the function when 0 (the y- intercept) To find the roots: Set f() 0 and solve the equation: f() 2-3, /2 f() f() Root (1.5) *Can be determined analytically -10

13 Finding the roots of a quadratic equation f() a 2 +b + c f() : Some quadratic function of variable a, b, c : Coefficients To find the roots: Set f() 0 and solve the equation: Use the quadratic formula: * Can be determined analytically f() 30 b ± b 2 4ac 2a y ± ( 3) 2 4*2*( 1) 2*2 3± and 1.78

14 quadratic function is a polynomial function with degree 2 f() a 2 +b + c Polynomials of degree n are of the general form f() a n n +a n-1 n-1 + a 2 2 +a 1 +a 0 Polynomial roots cannot always be determined analytically, but it is trivial in MTLB In MTLB Find the roots of polynomial p p Step 1: Set p 0, put in vector form p [3,8,0,-2,6] Step 2: Use root command roots (p) returns roots of polynomial p

15 Finding the roots of a nonlinear equation Given the a non-linear equation, g ( ) f ( ) We can rewrite it in the general form of a root-finding problem as g ( ) f ( ) or 0 F( ) 0 where F() g( ) f ( ).

16 F ( ) F ' ( i ) F ( i ) F ( ) 0 i i+ 1 F ' ( i ) F ( i i 1 )( ) i Make an initial guess for the root of the equation ( i ) The iterative Newton s Method follows as i + 1 i F( i F ( i ) )

17 In general, nonlinear equations can have more than 1 root the iterative Newton s method allows you to approimate 1 root that is close to your initial guess. This is termed a local root. Eample 3 ln() + 3 F() 3 - ln () 3 F () 3 2 1/ i+ 1 i F( i F ( i ) ) Iteration X approimation F() F () Refinement F(1.52) 0.09 Fairly close to 0 after only 3 iterations

18 Finding the roots of a nonlinear equation in MTLB We want to use the command: fzero fzero ( function name, initial guess) returns the root of the function that is closest to your initial guess Function: F() 3 - ln () 3 Initial guess for local root: 1 To use this command, we must first write a separate function file that specifies the function of interest In MTLB Go over eample of using fzero to find the local root of a nonlinear function