Numerical Methods in Physics

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1 Numerical Methods in Physics Numerische Methoden in der Physik, Instructor: Ass. Prof. Dr. Lilia Boeri Room: PH Tel: Address: Room: TDK Seminarraum Time: 8:30-10 a.m. Exercises: Computer Room, PH EG 004 F hpp://itp.tugraz.at/lv/boeri/num_meth/index.html (Lecture slides, Script, Exercises, etc.

2 TOPICS (this year: Chapter 1: Introduc8on. Chapter 2: Numerical methods for the solu8on of linear inhomogeneous systems. Chapter 3: Interpola\on of point sets. Chapter 4: Least- Squares Approxima8on. Chapter 5: Numerical solu8on of transcendental equa8ons. Chapter 6: Numerical Integra\on. Chapter 7: Eigenvalues and Eigenvectors of real matrices. Chapter 8: Numerical Methods for the solu8on of ordinary differen8al equa8ons: ini8al value problems. Chapter 9: Numerical Methods for the solu\on of ordinary differen\al equa\ons: marginal value problems.

3 Last week(29/10/2013 Least Square Approxima8on: Defini\on of the problem. Sta8s8cal distribu\on of experimental data: normal and Poisson distribu\on. Sta8s8cal proper\es of the fiqng parameters. Model Func\ons with Linear parameters. How is this implemented in prac8ce?

4 Least- Squares Approxima8on: Mathema8cal Formula8on: Given a set of n points (x i /y i, we wish to find a curve f(x which approximates the points as closely as possible taking into account possible uncertaini\es due to measurement errors. We also would like to be able to assign different weights to the points through suitable weigh\ng factors. n [ ] 2 χ 2 = g k y k f ;a min y k χ 2 g k > 0 f ;a a = a 1,..., a q Experimental values Weighted error sum Weigh8ng factors (sta8s8cal Model func8on Model (fiqng parameters

$Sta8s8cs in the LSQ method: n χ 2 = g k y k f ;a [ ] 2 min The weigh8ng factors g k depend on the sta\s\cs of the experimental data sets y k : Normal$

5 Sta8s8cs in the LSQ method: n χ 2 = g k y k f ;a [ ] 2 min The weigh8ng factors g k depend on the sta\s\cs of the experimental data sets y k : Normal distribu8on (analogic g k = 1 σ k 2 P(x;µ,σ = 1 # (x µ exp% 2πσ $ 2σ 2 2 ( ' g k = 1 y k Poisson distribu8on (digital P pois (X = k = f (k;λ = λ k e λ k!

6 Normal matrix of the LSQ fit: For the weighted error sum χ 2 we can define a normal matrix as: [ N] ij = χ 2 α i α j Its inverse is the covariance matrix, which gives important informa\on on the sta\s\cal proper\es of the fifng parameters: C = N 1 Covariance Matrix σ ai = c ii Standard devia8on of the a i s (fifng parameters r ij = c ij c ii c jj Correla8on coefficients for the a i s: ( cij 1 Measure how strongly the i- th and j- th parameter influence each other.

7 Model func8ons with linear parameters: A model func\on with linear parameters has the form: m f (x;a = a j ϕ j (x Here, φ j (x are arbitrary (linearly independent basis func8ons. f(x,a is linear in the fi0ng parameters, not in x. The formula for χ 2 can be recast into a linear inhomogeneous problem: Âa = β j=1 n Â "# α ij $ % α ij = g k ϕ i ϕ j β i = g k y k ϕ i It can be shown that A also coincides with the normal matrix of the problem. n

8 How to implement LSQ with linear model parameters: 1 Input the experimental data set: x k, y k and the sta\s\cal weights g k 2 Choose a set of basis func8ons φ j (x: 3 Construct the auxiliary linear problem: n f (x;a = a j ϕ j (x Âa = β Â "# α ij $ % α ij = g k ϕ i ϕ j, β i = g k y k ϕ i 4 Solve the linear problem (LU decomposi\on; Find op\mal fifng parameters. a opt = (a opt 1,..., a opt q 5 Calculate the covariance matrix (standard devia\ons of the fifng parameters. m j=1 n 6 Calculate the value of the op\mal fifng func\on on the given data points: f ;a opt 7 Evaluate the weighted error sum. n [ ] 2 χ 2 = g k y k f ;a

9 This week(5/11/2013 Least Square Approxima8on in prac8ce. Descrip\on of a program to perform the LSQA for model func\ons with linear parameters. Model func\ons with non- linear parameters: defini\on. Model func\ons with non- linear parameters: easy lineariza\on tricks.

10 How to implement LSQ with linear model parameters: 1 Input the experimental data set: x k, y k and the sta\s\cal weights g k 2 Choose a set of basis func8ons φ j (x: 3 Construct the auxiliary linear problem: n f (x;a = a j ϕ j (x Âa = β Â "# α ij $ % α ij = g k ϕ i ϕ j, β i = g k y k ϕ i 4 Solve the linear problem (LU decomposi\on; Find op\mal fifng parameters. a opt = (a opt 1,..., a opt q 5 Calculate the covariance matrix (standard devia\ons of the fifng parameters. m j=1 n 6 Calculate the value of the op\mal fifng func\on on the given data points: f ;a opt 7 Evaluate the weighted error sum. n [ ] 2 χ 2 = g k y k f ;a

11 Program LFIT: input and output parameters INPUT: X,Y: vectors that contain the x k, y k for the LSQ fit. SIG: vector with the standard devia\ons for the data (σ k. NDATA: Number of data points. MA: Number of parameters in the model func\on (=q. OUTPUT: A: Vector which contains the op\mized parameters for the model func\on (a opt. YF: Vector which contains the values of the op\mized fifng parametrs in x k - f. COVAR: Covariance matrix. CHISQ: Value of the weighted error sum for a=a opt.

12 Program LFIT: structure of the program 1 Calcula\on of the matrix elements α ij and of the components of the inhomogeneous vector β i of the inhomogeneous vector. These quan\\es are stored in NORMAL(, and BETA. 2 Calcula\on of the op\mised fifng parameters and of the covariance matrix through the rou\nes LUDCMP and LUBKSB (LU decomposi\on. 3 Calcula\on of the weighted error sum for the best- fit curve using the input x k, y k and the op\mised fifng parameters. External Func8ons/Rou8nes: FUNCS (Contains the basis func\ons used for the op\mal fit. LUDCMP and LUBKSB: rou\nes used for the LU decomposi\on and for the inversion of the normal matrix - > covariance matrix.

13 Ini8alize arrays to zero: normal (A and beta (β. Evaluate the components of A and beta: solve A x=β - > Get op8mal values of the fiqng parameters. Use the op8mal values of the fiqng parameters to evaluate χ 2 Calculate Covariance matrix: C=A - 1.

14 Evaluate the components of A and beta. solve A x=β - > Get op8mal values of the fiqng parameters.

15 n Â "# α ij $ % α ij = g k ϕ i ϕ j β i = g k y k ϕ i Iterate on,n k (NDATA n Evaluate φ i External call to funcs. Note that there are q=ma basis func\ons. ϕ i afunc(i Evaluate the sta\s\cal weights g k =(1/σ k 2 g k > g

16 n Â "# α ij $ % α ij = g k ϕ i ϕ j β i = g k y k ϕ i (Iterate on,n k (NDATA n Iterate on i=1,q (NA index for basis func\ons (rows. Iterate on j=1,i (NA index for basis func\ons (columns. The A matrix and β vector are constructed row by row for each x k. α ij (k = g k ϕ i ϕ j normal(i,j β i (k = g k y k ϕ i beta(i Up to here. f (k new=old + new k

17 n Â "# α ij $ % α ij = g k ϕ i ϕ j β i = g k y k ϕ i Fill up the A (normal matrix, which has to be symmetric: n j=1 j=2 j=3 j=4 i=1 i=2 i=3 i=4 i=2, q j=1,i- 1 We fill the lower half!

18 Solve the auxiliary problem: Âa = β Using LU decomposi8on (see lect. 3, 15/10/2013 and script, chapter 2. LUDCMP: The matrix NORMAL is decomposed into lower + upper triangular matrices; NORMAL contains now the LU decomposi8on of A. LUBKSB: The linear inhomogeneous problem is solved with backward + forward recursion. BETA is the inhomogeneous vector, LOES is the solu\on. a = a opt = (a 1 opt,..., a q opt LOES

19 The q (MA op8mal parameters are saved into the vector A(j. Use the op8mal values of the fiqng parameters to evaluate χ 2 n [ ] 2 χ 2 = g k y k f ;a

20 Op8mal fiqng func8on: f (x;a opt = q j=1 a j opt ϕ j (x Evaluate φ i External call to funcs. ϕ i afunc(i Evaluate f j f j ;a opt sum f j ;a opt = a j opt ϕ j

21 Iterate over j: f ;a opt opt = a j ϕ j = f j χ k 2 = g k q j=1 f ;a opt = SUM YF(k [ y k f ;a] 2 Close the itera8on on k: k χ 2 = χ k 2 q j=1

22 Calculate the covariance matrix: C = N 1 Using LU decomposi8on (see lect. 3, 15/10/2013 and script, chapter 2.

23 Possible uses: 2 Inversion of a matrix: A X = I, X A 1 A x x n1 $ % ' ( = $ % ' (... A x 1n... x nn $ % ' ( = $ % ' (

24 A X = I, X A 1 The inverse matrix can be constructed as a collec\on of n column vectors:! # # # " x x n1 $! #...# # % " x 1 j... x nj $! #...# # % " x 1n... x nn $ % The same is true for the iden8ty matrix, where the column vectors are:! # # # " $! #...# # % " 0 1 x nj $! #...# # % " $ % Only the j th element is non- zero and equal to one.

25 A X = I, X A 1 The original problem reduces to n equivalent vector problems: A x j = b j For the column vectors x j and b j. In this way the LU decomposi\on of A is performed only once, and one has to solve (LUBKSB n \mes an inhomogeneous system, with a different b j.

26 Run over columns: Construct the inhomogeneous vector for the j- th column: β j Solve the linear system Ax j =β j The column vector x j is saved into the covariance matrix.

27 How to use LFIT? Solving a LSQ problem in prac\ce

28 Given a data set, we want to obtain the best fit with LSQ, and evaluate the quality of the fit (variance and standard devia\on.

29 Normal matrix of the LSQ fit: For the weighted error sum χ 2 we can define a normal matrix as: [ N] ij = χ 2 α i α j Its inverse is the covariance matrix, which gives important informa\on on the sta\s\cal proper\es of the fifng parameters: C = N 1 Covariance Matrix σ ai = c ii Standard devia8on of the a i s (fifng parameters r ij = c ij c ii c jj Correla8on coefficients for the a i s: ( cij 1 Measure how strongly the i- th and j- th parameter influence each other.

30 Variance and Standard Devia8on: V = χ 2 N q Variance q σ V = N q 2 N q Standard Devia8on Number of fiqng parameters Number of degrees of freedom Quality of the LSQ fit: In case of an ideal model for N>>1, V has approximately a normal distribu8on with E=1 and σ=σ V. If V lies significantly outside the interval: [1- σ V,1+σ V ] the fit is bad!

31 A prac8cal example: Let us imagine that we want to find the best fifng func\on for N k =101 points, randomly generated around a polynomial curve, with a given (constant standard devia\on σ. The (third degree polynomial is: y(x = 0.5 x 0.2x x 3 The op\mal fit would be given by the linear model func\on (q=4: With: f (x, a = 4 j=1 a j x j 1 a 0 = 0.5 a 1 = 1 a 2 = 0.2 a 3 = 0.1

32 First data set (σ=0.1 a 0 = 0.5 a 1 = 1 a 2 = 0.2 a 3 = 0.1 Real a 0 = a 1 = a 2 = a 3 = Fit

33 σ V = 2 N q V ±σ V =

34 Second data set (σ=0.025, improved sta\s\cs:

35 σ V = 2 N q V ±σ V =

36 Functions with non- linear parameters Lineariza\on method (special cases + Gauss Newton method

37 Model func8ons with non- linear parameters: If we assume as fifng func\on an exponen8al: f (x;a, b = a e bx We obtain for the weighted error sum: χ 2 = n g k $ y k a be bx k % The minimum condi8on gives: ' Min! $ % ' n a g k e 2bx k = g k y k e bx k n n a g k e 2bx k = g k x k y k e bx k n The system of equa8ons is non- linear!

38 However, the exponen\al func\on is usually linearized using logarithms: ln( f (x;a, b = ln(a e bx = ln(a b x = ln(y k

39 The set of equa\ons for the a,b parameters is also linear: " $ $ $ $ # n x k k 2 x k x k k k % ' " ' ' $ # ' ln(a b " % $ ln(y k ' = $ k $ x k ln(y k $ # k % ' ' ' ' The solu\on is: $ ( / D a = exp % ln y k x 2 k x k x k ln y k ( / D b = n x k ln y k x k ln y k ( D = n x k 2 x k 2 '

$Remark: hidden correla8ons Hidden correla\ons between the fifng parameters can occur when the parameters$

40 Remark: hidden correla8ons Hidden correla\ons between the fifng parameters can occur when the parameters chosen to represent a func\on are not really independent. f (x;a, b,c = a e bx+c f (x, a' = a e c e bx = a'e bx

41 This week(5/11/2013 Least Square Approxima8on in prac8ce. Descrip\on of a program to perform the LSQA for model func\ons with linear parameters. Model func\ons with non- linear parameters: defini\on. Model func\ons with non- linear parameters: easy lineariza\on tricks.

Numerical Methods in Physics

Numerical Methods in Physics Numerical Methods i Physics Numerische Methode i der Physik, 515.41. Istructor: Ass. Prof. Dr. Lilia Boeri Room: PH 03 090 Tel: +43-316- 873 8191 Email Address: l.boeri@tugraz.at Room: TDK Semiarraum Time: