Linear Systems of Equations. ChEn 2450

Transcription

1 Linear Systems of Equations ChEn 450 LinearSystems-directkey - August 5, 04

2 Example Circuit analysis (also used in heat transfer) + v _ R R4 I I I3 R R5 R3 Kirchoff s Laws give the following equations for the currents (I) in terms of the resistances (R) v R I R 4 I 4 = 0 R I + R I + R 3 I 3 = 0 R 4 I 4 R 3 I 3 + R 5 I 5 = 0 I 6 = I + I I + I 3 = I 4 I = I 3 + I 5 I6 I4 I5 Assuming the voltage (v) and resistances (R i ) are known, solve for the currents, I j Define the ordering for the unknowns (x) Define the A matrix 3 Define the b (rhs) vector 4 Solve for x LinearSystems-directkey - August 5, 04

3 Systems of Linear Equations Any system of linear equations may be written as: a x + a x + + a n x n = b a x + a x + + a n x n = b a a a n a a a n x x = b b a m x + a m x + + a mn x n = b m a m a m a mn to be well-posed, what is the relationship between m and n? x n Unknowns b n Alternatively, Ax=b, or n j= a i,j x j = b i Note: in MATLAB, you may easily solve a system of linear equations by defining a square matrix, A, a column vector, b, and setting x=a\b; Example: 5 equations Define the A matrix (here we assume that a ij have been defined previously, or substitute appropriate numbers) Define the b (rhs) vector 3 Solve for x A = [ a a a3 a4 a5;! a a a3 a4 a5;! a3 a3 a33 a34 a35;! a4 a4 a43 a44 a45;! a5 a5 a53 a54 a55; ];!! b = [ b b b3 b4 b5 ]';!! x = A\b; 3 LinearSystems-directkey - August 5, 04

4 Hoffman 3-4 Direct Solution Methods Gaussian Elimination Algorithm to solve systems of linear equations Basically a methodic approach to solving equations by hand Cost scales as n 3 where n is the number of equations LU Factorization Useful when you need to solve Ax=b for different b but same A L - Lower diagonal matrix, U - Upper diagonal matrix Determining L, U is expensive, but solving the resulting system is simple See help lu in MATLAB a a a m a a a m a n a n a nm A = LU = n n,m L b = b LUx = b Ux = b NOTE: if you are rusty on solution of linear systems, read through Hoffman -3 carefully & do the examples! u u u m 0 u u m u nm 4 LinearSystems-directkey - August 5, 04

5 Hoffman 7 Sparse Linear Systems Sparse systems of equations have mostly zeros in the matrix Often arise from solving ODEs or PDEs Also in networked systems Networked flow systems - separation units (you will see this in later ChEn courses) Networked spring systems Electrical circuits many others Idea: instead of storing all of the zeros, only store the nonzero entries, along with the row & column that they belong in For k non-zeros per equation, this reduces storage from n to kn For small n, storing all of the zeros isn t a problem For large n, it is Typically solved by iterative methods 5 LinearSystems-directkey - August 5, 04

6 Sparse Matrices Example: Steady-state, -dimensional heat transfer These are tridiagonal matrices (special kinds of sparse matrices) d T dx = S solve at a discrete set of points: more on how to do this later in the semester Discrete solution for n=5 points x T (0) = T o Discrete solution for n=7 points x x=0 x=l T T T 3 T 4 T 5 = T T T 3 T 4 T 5 T 6 T 7 S x T o S S 3 S 4 S 5 = S x T o S S 3 S 4 S 5 S 6 S 7 dt dx x=l = 0 How would you write a MATLAB code to generate the matrix & rhs vector (given n)? 6 LinearSystems-directkey - August 5, 04

7 MATLAB Tip: Creating Banded Matrices diag(v,k) - create a matrix with the vector v on its k th diagonal k = 0 main diagonal k = upper diagonal k = - lower diagonal k = nd upper diagonal d = [ ];! ud = [ ];! ld = [ ];!! A = diag(ld,-) + diag(d,0) + diag(ud,); A = n = 5;! d = -*ones(n,);! ud = ones(n-,);! ld = ones(n-,);!! A = diag(ld,-) + diag(d,0) + diag(ud,); A = This creates a dense matrix If you get into large systems of equations, you will need to use sparse matrices See the sparse function in MATLAB 7 LinearSystems-directkey - August 5, 04

8 Hoffman 5 Tridiagonal Matrices ( Special Kinds of Sparse Matrices) Tx = b T = = a a a a a 3 d u d u a n,n a n,n a n,n a n,n a n,n n d n u n n d n NOTE: there are many different varieties of the Thomas algorithm They are all equivalent No need to store n entries of T - there are only 3n- non-zero entries lower diagonal (n- entries) d main diagonal (n entries) u upper diagonal (n- entries) The Thomas Algorithm: (Given, u, d, b) for i=,3,,n d i = d i u i i d i b i = b i b i i d i x n = b n dn, x i = b i u i x i+ d i i = n,, 8 LinearSystems-directkey - August 5, 04

9 Thomas Algorithm - Example x 3x = 4 x + x x 3 = 4x x 3 + x 4 = 9 x 3 x 4 = 3 4 x x x 3 x 4 = 4 9 = 4 d = u = 3 Set d i = d i u i i d i, b i = b i b i i d i i =, 3,, n d = d u d = d 3 = d 3 u d = d 4 = d 4 u 3 3 d 3 = ( 3)() = 7 ( )(4) = 7/ 7 ()() /7 = 5 x n = b n dn, i = n, x i = b i u i x i+ d i i = n,, x 4 = b 4 d 4 = 60 5 = 4 x 3 = b 3 u 3 x 4 d 3 = 3/7 ()(4) /7 = 3 x = b u x 3 = 4 ( )(3) = d 7/ x = b u x = 4 ( 3)() = d b = b b d = b 3 = b 3 b d = 9 b 4 = b 4 b 3 3 d 3 = Recall: d and b remain unchanged ( 4)() = 4 (4)(4) 7/ = 3 7 (3/7)() = 60 /7 What kinds of error are present? Iteration error? Roundoff error? Approximation error? 9 LinearSystems-directkey - August 5, 04