Numerical Methods in Physics

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1 Numerical Methods i Physics Numerische Methode i der Physik, Istructor: Ass. Prof. Dr. Lilia Boeri Room: PH Tel: Address: l.boeri@tugraz.at Room: TDK Semiarraum Time: 8:30-10 a.m. Exercises: Computer Room, PH EG 004 F hpp://itp.tugraz.at/lv/boeri/num_meth/idex.html (Lecture slides, Script, Exercises, etc).

2 TOPICS (this year): Chapter 1: Itroduc8o. Chapter : Numerical methods for the solu8o of liear ihomogeeous systems. Chapter 3: Iterpola\o of poit sets. Chapter 4: Least- Squares Approxima8o. Chapter 5: Numerical solu8o of trascedetal equa8os. Chapter 6: Numerical Itegra\o. Chapter 7: Eigevalues ad Eigevectors of real matrices. Chapter 8: Numerical Methods for the solu8o of ordiary differe8al equa8os: ii8al value problems. Chapter 9: Numerical Methods for the solu\o of ordiary differe\al equa\os: margial value problems.

3 Last week(/10/013) Liear Systems: Direct (LU decomposi8o) vs idirect methods (Gauss- Seidel). Direct methods, itera8ve improvemet of the solu\o (reduce roudoff). Direct methods, special cases: tridiagoal matrices. Idirect methods: itera8ve solu\o for sparse matrices. Idirect methods: Gauss- Seidel itera8o rule. Bad matrices: defii\o ad proper\es. Gauss- Seidel itera\o for bad matrices, storig iforma\o ad efficiecy. Gauss- Seidel method with over ad uder- relaxa8o.

4 Liear Systems Methods of Solu8o (umerical): Âx = b Direct Methods: No methodological error, BUT computa\oally expesive; roudoff errors ca be large. LU decomposi8o. Itera8ve Methods: Simple algorithms; roudoff is easily cotrolled. The solu8o is approximate (truca8o). Gauss- Seidel method.

$Direct Methods: Âx = b ˆ Ux = y Two- steps procedure: 1) LU decomposi8o (Dooli;le ad Crout): Reformula\o of the Gaussia elimia\o.$

5 Direct Methods: Âx = b ˆ Ux = y Two- steps procedure: 1) LU decomposi8o (Dooli;le ad Crout): Reformula\o of the Gaussia elimia\o. A real matrix ca always be represeted as the product of two real triagular matrices L ad U, i.e. Â = ˆL ˆ U ) The two auxiliary systems are solved through back ad forward subs\tu\o: ˆ U x = y Back ˆL y = b Forward

6 Gauss- Seidel Method: itera8ve method for liear ihomogeeous sets of equa\os. Advatages: simplicity; the matrix of coefficiets is ot chaged durig the itera\o. Very efficiet for systems with a sparse matrix of coefficiets. x i (t+1) = x i (t) Δx i (t) Δx i (t) = x i (t) + 1 a ii % ' &' j=1( j i) (i =1,..., ) ( a ij x (t) j b i * )* Star\g from a ii\al vector x 0 (star%g vector) oe obtais a sequece of vectors x (t), which coverges to the exact solu\o: lim x (t) x t I prac\ce, the itera\o stops whe a give precisio is reached: xi(t)- xi(t- 1) ε For sparse matrices, oly a few terms survive i the equa\o for the Δx i s.

7 Efficiecy of direct Methods: LU GS Efficiecy of the G- S method compared to LU decomposi\o for a Laplace equa\o (sparse matrix).

8 This week(9/10/013) Least Square Approxima8o: Defii\o of the problem. Sta8s8cal distribu\o of experimetal data: ormal ad Poisso distribu\o. Sta8s8cal proper\es of the fi[g parameters. Model Fuc\os with Liear parameters. How is this implemeted i prac8ce?

9 Least Squares Approximatio Op\mal Fikg of a data set

10 Least- Squares Approxima8o: Basic problem: Data collected from experimets form a discrete set. Physical processes are typically described by mathema\cal expressios, which employ real umbers. These are easily maipulated with the stadard tools of algebra (itegrals, deriva\ves, etc). How to reduce a discrete set to a co8uous fuc8o? Iterpola8o: Fid a aaly\cal expressio (polyomial) which passes exactly through all experimetal poits. Least- squares approxima8o: Fid the best possible aaly\cal formula to approximate the behaviour of the experimetal poits.

11 Example: We wish to measure the resis\vity of a metallic coductor: The data poits (Voltage: V; Curret I) follow Ohm s law, but are affected by a experimetal error. V = R I

12 R Due to experimetal errors, the data do ot lie exactly o top of a straight lie (Ohm s law), but oly approximately.

13 Least- Squares Approxima8o: Mathema8cal Formula8o: Give a set of poits (x i /y i ), we wish to fid a curve f(x) which approximates the poits as closely as possible takig ito accout possible ucertaii\es due to measuremet errors. We also would like to be able to assig differet weights to the poits through suitable weigh\g factors. [ ] χ = g k y k f (x k ;a) k=1 mi y k χ g k > 0 f (x k ;a) a = a 1,..., a q Experimetal values Weighted error sum Weigh8g factors (sta8s8cal) Model fuc8o Model (fi[g) parameters

14 Sta8s8cs: The quality of the results of a LSQ fit depeds o the sta\s\cal quality of the data to be fiped (y k ). The sta\s\cs of the y k s is defied i terms of their expecta\o value (E k ) ad stadard devia\o (σ k ). j E k = 1 N j =1,..., N N ( j) y k j=1 Expecta8o value Number of measuremets

15 Stadard Devia8o (σ k ): measures the spread of the y k s aroud their expecta\o value E k. # % σ k = % % % $ N j=1 ( ( y j) k E k ) & ( ( N 1 ( ( ' 1/ spread

16 Gaussia ormal distribu8o: It is useful to describe experimetal aalogue measuremets. P(x;µ,σ ) = 1 # (x µ) exp% πσ $ σ & ( ' Parameters: µ Media of the distribu\o σ Stadard Devia8o σ Variace Cetral limit theorem: the mea of a radom variable is distributed approximately ormally, irrespec\ve of the form of the origial distribu\o - > very commo!!! Stadard ormal distribu8o: μ=0, σ=0.

17 Normalized Gaussia distribu8o: We will assume that our measured (y) values have a ormal distribu\o, with media = expecta\o value, ad stadard devia\o=exp. stadard devia\o. P(y E) = 1 # (y E) exp% πσ $ σ & ( ' P(y- E) describes the probability with which a specific distace (y- E) betwee the measured value y ad its expecta\o value occurs. The stadard devia\o idicates the spread with which the measured values are distributed aroud their expecta\o value E. Data- sets with a smaller σ are more accurate.

18 P(y E) = 1 # (y E) exp% πσ $ σ & ( ' y = E ±σ Iflec\o poits (P (y)=0).

19 Geometrical Meaig of the Stadard Devia8o: +σ σ +σ σ +3σ 3σ d(y E) d(y E) d(y E) 1 $ (y E) exp& πσ % σ 1 $ (y E) exp& πσ % σ 1 $ (y E) exp& πσ % σ ' ) = ( ' ) = ( ' ) = ( σ σ 3σ

20 Poisso s sta8s8cs: The Poisso s distribu\o is a discrete probability distribu8o that expresses the probability of a give umber of evets occurrig i a fixed iterval of \me ad/or space if these evets occur with a kow average rate ad idepedetly of the \me sice the last evet. Proper8es: P pois (X = k) = f (k;λ) = λ k e λ k! E(X) = λ var(x) = λ Expecta8o Value Variace I a Poisso s distribu\o the expecta\o value ad variace are equal.

21 The Poisso s distribu\o ca be applied to systems with a large umber of possible evets, each of which is rare. Example: Cou\g Experimet (radioac\ve decay): D is a digital couter; the umber of couts has a Poisso distribu\o.

$Sta8s8cs i the LSQ method: k=1 χ = g k y k f (x k ;a) [ ] mi The weigh8g factors g k deped o the sta\s\cs of$

22 Sta8s8cs i the LSQ method: k=1 χ = g k y k f (x k ;a) [ ] mi The weigh8g factors g k deped o the sta\s\cs of the experimetal data sets y k : g k = 1 σ k g k = 1 y k Normal distribu8o (aalogic) Poisso distribu8o (digital)

23 Normal matrix of the LSQ fit: For the weighted error sum χ we ca defie a ormal matrix as: [ N] ij = 1 χ α i α j Its iverse is the covariace matrix, which gives importat iforma\o o the sta\s\cal proper\es of the fikg parameters: C = N 1 Covariace Matrix σ ai = c ii Stadard devia8o of the a i s (fikg parameters) r ij = c ij c ii c jj Correla8o coefficiets for the a i s: ( cij 1) Measure how strogly the i- th ad j- th parameter ifluece each other.

Variace ad Stadard Devia8o: V = χ N q Variace q σ V = N q N q Stadard Devia8o Number of fi[g parameters Number of degrees of freedom Quality of the LSQ fit: I

24 Variace ad Stadard Devia8o: V = χ N q Variace q σ V = N q N q Stadard Devia8o Number of fi[g parameters Number of degrees of freedom Quality of the LSQ fit: I case of a ideal model for N>>1, V has approximately a ormal distribu8o with E=1 ad σ=σ V. If V lies sigificatly outside the iterval: [1- σ V,1+σ V ] the fit is bad!

25 Model Fuctios with Liear Parameters

26 Model fuc8os with liear parameters: A model fuc\o with liear parameters has the form: m f (x;a) = a j ϕ j (x) j=1 Here, φ j (x) are arbitrary (liearly idepedet) basis fuc8os. f(x,a) is liear i the fikg parameters, ot i x. I the special case i which oe uses as basis fuc\o a straight lie: φ j (x)=f li (x; a 1,a ) we have the liear regressio. Liear Regressio: f (x;a 1, a ) = a 1 + a x

27 Model fuc8os with liear parameters, LSQ formula: Iser\g the expressio of the model fuc\o with liear parameters i the expressio for the weighted error sum we have: # m & χ = g k % y k a j ϕ j (x k )( k=1 $ % j=1 '( Derivig with respect to the fikg parameters we get: mi χ $ m ' = g k ϕ i (x k )& y k a j ϕ j (x k )) = 0 i =1,..., m a i k=1 %& j=1 () Which ca be recast i liear set of m equa\os: m a j g k ϕ i (x k )ϕ j (x k ) j=1 k=1 = g k y k ϕ i (x k ) Âα = β k=1

28 This ihomogeeous liear set of m equa\os is (i matrix form): Âa = β Â "# α ij $ % α ij = g k ϕ i (x k ) ϕ j(x k ) k=1 β i = g k y k ϕ i (x k ) k=1 Oe ca defie a ormal matrix, give by: ˆN "# ij $ % ij = 1 χ = g k ϕ i (x k ) ϕ j(x k ) = α ij a i a j k=1

29 Stadard Devia8o of the fi[g parameters: We cosider the liear regressio formula (q=): f (x;a 1, a ) = a 1 + a x The least- squares formula gives: [ ] χ = g k y k a bx k k=1 mi a = a opt, b = b opt The ormal matrix of the problem is give by: with: α 11 =! N = α 11 α $ 1 # " α 1 α & % g k α 1 = g k x k k=1, k=1, α = g k x k k=1,

30 The op%mized parameters (a opt ad b opt ) ca be obtaied from:! N # " a b $! & = % # " β 1 β $ & % with: β 1 = g k y k, β = g k x k y k k=1 This gives: k=1 a = a opt = β 1α β α 1 D, b = b opt = β α 11 β 1 α 1 D D is the determiat of the ormal matrix: D = α 11 α α 1

31 The covariace matrix is the iverse of the ormal matrix: C = N 1 = 1 " D $ # α α 1 α 1 α 11 % ' & The diagoal elemets give the stadard devia\os of the fikg parameters: σ a = α D, σ b = α 11 D a = a opt ±σ a, b = b opt ±σ b We will ow show that the same result ca be obtaied usig the Error Propaga8o Rule (EPR).

32 Error Propaga8o Rule: The fikg parameters are fuc\o of the x, y compoets of the data poits: a = a(x 1,..., x ; y 1,..., y ) b = b(x 1,..., x ; y 1,..., y ) We ca the use the stadard formula for the error propaga8o. Give a fuc\o f(x) of variables (x 1,,x ), with errors Δx 1,, Δx the error o the fuc8o is give by: f = f (x 1,..., x ) = f (x) f (x) = f (x)± Δf (x) ( Δf (x)) = )# f & + % ( * + $ x 1 ' # (Δx 1 ) % f $ x & ( ', (Δx ). -.

33 Applyig this to the expressio for the stadard devia\os gives: σ a = σ b = l=1 l=1 (" * a $ )* # x l (" * b $ )* # x l % ' & % ' & " σ (x l ) + $ a # y l " σ (x l ) + $ b # y l % ' & % ' & + σ (y l ) -,- + σ (y l ) -,- If we assume that there is o sta\s\cal error o the x l s, ad that the y l s have a ormal distribu8o, we have: Ad thus: 1 σ (x l ) = 0 g l = σ (y l ) ( σ 1 " a % + a = * $ ' - )* g l # y l &,- ; σ b = l=1 l=1 ( * 1 )* g l " b $ # y l % ' & + -,-

34 To calculate the par8al deriva8ves we have to recall: a = a opt = β 1α β α 1 D, b = b opt = β α 11 β 1 α 1 D α 11 = g k,α 1 = g k x k k=1, k=1,,α = g k x k k=1, α ij y l = 0; β 1 = g k y k, β = g k x k y k k=1 k=1 β 1 y l = g l ; β y l = g l x l Subs\tu\g i the expressios for the stadard devia\os we get: σ a = σ b = l=1 l=1 1 " 1 g l D (α % g l α 1 g l x l ) # $ & ' 1 " 1 g l D ( α % 1g l +α 11 g l x l ) # $ & '

35 We obtai (for example for a): σ a = l=1 1 " 1 g l D (α % g l α 1 g l x l ) # $ & ' 1 ) 1 = g l D g, + l. [(α α 1 x l )] = * - l=1 = 1 g D l [(α α 1 x l )] = l=1 = 1 g D l g k g k' " # x k x k' x l x k x k' + x l x k x k' % & l,k,k' Reamig the idexes of the sum i the last two terms we obtai: σ a = 1 g D l g k g k' x k # $ l,k,k' x k' = x l x k' % & = 1 # g D k x k - k=1 $ - g l g k' l=1 k'=1 x ' g x * k' ), ( l l + l=1 %. &.

36 σ a = 1 ) g D k x k + k=1 * + g l g k' l=1 k'=1 x # g x & k' % ( $ l l ' l=1,. -. D = α 11 α α 1 α 11 = g k,α 1 = g k x k k=1, k=1,,α = g k x k k=1, We thus obtai: σ a = 1 D D g kx k = α D k=1 i.e. the diagoal terms of the covariace matrix give the stadard devia\o for the fikg parameters!!!!

37 I summary (LSQ with liear model parameters): 1) Iput the experimetal data set: x k, y k ad the sta\s\cal weights g k ) Choose a set of basis fuc8os φ j (x): 3) Costruct the auxiliary liear problem: f (x;a) = a j ϕ j (x) Âa = β Â "# α ij $ % α ij = g k ϕ i (x k ) ϕ j(x k ), β i = g k y k ϕ i (x k ) k=1 4) Solve the liear problem (LU decomposi\o); Fid op\mal fikg parameters. a opt = (a opt 1,..., a opt q ) 5) Calculate the covariace matrix (stadard devia\os of the fikg parameters). m j=1 k=1 6) Calculate the value of the op\mal fikg fuc\o o the give data poits: f (x k ;a opt ) 7) Evaluate the weighted error sum. [ ] χ = g k y k f (x k ;a) k=1

38 This week(9/10/013) Least Square Approxima8o: Defii\o of the problem. Sta8s8cal distribu\o of experimetal data: ormal ad Poisso distribu\o. Sta8s8cal proper\es of the fi[g parameters. Model Fuc\os with Liear parameters. How is this implemeted i prac8ce?

Numerical Methods in Physics

Numerical Methods in Physics Numerische Methoden in der Physik, 515.421. Instructor: Ass. Prof. Dr. Lilia Boeri Room: PH 03 090 Tel: +43-316- 873 8191 Email Address: l.boeri@tugraz.at Room: TDK Seminarraum