Convergence Criteria

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Convergence Criteria

Convergence Criteria

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  • 1. Convergence Criteria 1
  • 2. Iterative Solution Methods    Starts with an initial approximation for the solution vector (x0) At each iteration updates the x vector by using the sytem Ax=b During the iterations A, matrix is not changed so sparcity is preserved  Each iteration involves a matrix-vector product  If A is sparse this product is efficiently done 2
  • 3. Iterative solution procedure     Write the system Ax=b in an equivalent form x=Ex+f (like x=g(x) for fixed-point iteration) Starting with x0, generate a sequence of approximations {xk} iteratively by x k+1 =Ex k +f Representation of E and f depends on the type of the method used But for every method E and f are obtained from A and b, but in a different way 3
  • 4. Convergence     As k→∞, the sequence {xk} converges to the solution vector under some conditions on E matrix This imposes different conditions on A matrix for different methods For the same A matrix, one method may converge while the other may diverge Therefore for each method the relation between A and E should be found to decide on the convergence 4
  • 5. Different Iterative methods    Jacobi Iteration Gauss-Seidel Iteration Successive Over Relaxation (S.O.R)   SOR is a method used to accelerate the convergence Gauss-Seidel Iteration is a special case of SOR method 5
  • 6. Jacobi iteration a11 x1 + a12 x2 +  + a1n xn = b1 a21 x1 + a22 x2 +  + a2 n xn = b2  an1 x1 + an 2 x2 +  + ann xn = bn  x10   0 0  x2  x =   0  xn    1 0 0 1  (b1 − a12 x2 −  − a1n xn ) k +1 xi = bi − a11 aii  1 0 0 x1 = (b2 − a21 x10 − a23 x3 −  − a2 n xn ) 2 a22 1 0 0 x1 = (bn − an1 x10 − an 2 x2 −  − ann −1 xn −1 ) n ann 1 x1 =  ∑ aij x − j∑1aij x  j =1 =i +  i −1 k j n k j 6
  • 7. xk+1=Exk+f iteration for Jacobi method A can be written as A=L+D+U (not decomposition) 0 0  a11 0 0  0 a12 a13   a11 a12 a13   0 a a22 a23  =  a21 0 0 +  0 a22 0  + 0 0 a23   21         a31 a32 a33   a31 a32 0  0 0 a33  0 0 0         Ax=b ⇒ (L+D+U)x=b 1 k +1 xi = aii Dxk+1 i −1 n   k k bi − ∑ aij x j − ∑ aij x j  j =1 j = i +1    Lx k  Uxk Dxk+1 =-(L+U)xk+b xk+1=-D-1(L+U)xk+D-1b E=-D-1(L+U) f=D-1b 7
  • 8. Gauss-Seidel (GS) iteration Use the latest update a11 x1 + a12 x2 +  + a1n xn = b1 a21 x1 + a22 x2 +  + a2 n xn = b2  an1 x1 + an 2 x2 +  + ann xn = bn 1 0 0 1  (b1 − a12 x2 −  − a1n xn ) k +1 xi = bi − a11 aii  1 1 1 0 0 x2 = (b2 − a21 x1 − a23 x3 −  − a2 n xn ) a22 1 1 x1 = (bn − an1 x1 − an 2 x1 −  − ann −1 x1 −1 ) n 2 n ann 1 x1 =  x10   0 0  x2  x =   0  xn    i −1 ∑a x j =1 ij k +1 j  − ∑ aij x  j = i +1  n k j 8
  • 9. x(k+1)=Ex(k)+f iteration for Gauss-Seidel Ax=b ⇒ (L+D+U)x=b k +1 i x Dx 1 = aii k+1 i −1 n   k +1 k bi − ∑ aij x j − ∑ aij x j  j =1 j = i +1    Lx k +1  (D+L)xk+1 =-Uxk+b Uxk xk+1=-(D+L)-1Uxk+(D+L)-1b E=-(D+L)-1U f=-(D+L)-1b 9
  • 10. Comparison    Gauss-Seidel iteration converges more rapidly than the Jacobi iteration since it uses the latest updates But there are some cases that Jacobi iteration does converge but Gauss-Seidel does not To accelerate the Gauss-Seidel method even further, successive over relaxation method can be used 10
  • 11. Successive Over Relaxation Method  GS iteration can be also written as follows k +1 i x 1 =x + aii k i i −1 n   k +1 k bi − ∑ aij x j − ∑ aij x j  j =1 j =i   xik +1 = xik + δ ik Correction term ωδ i2 xi3 xi2 xi1 0 i x δ ωδ i1 2 i δ i1 δ i0 Multiply with ω >1 Faster convergence ωδ i0 11
  • 12. SOR xik +1 = xik + ωδ ik i −1 n   k +1 k x bi − ∑ aij x j − ∑ aij x j  j =1 j =i   i −1 n  1  k +1 k k +1 k xi = (1 − ω ) xi + ω bi − ∑ aij x j − ∑ aij x j  aii  j =1 j =i +1  k +1 i 1 = x +ω aii k i 1<ω<2 over relaxation (faster convergence) 0<ω<1 under relaxation (slower convergence) There is an optimum value for ω Find it by trial and error (usually around 1.6) 12
  • 13. x(k+1)=Ex(k)+f iteration for SOR 1 k +1 k xi = (1 − ω ) xi + ω aii i −1 n   k +1 k bi − ∑ aij x j − ∑ aij x j  j =1 j = i +1   Dxk+1=(1-ω)Dxk+ωb-ωLxk+1-ωUxk (D+ ωL)xk+1=[(1-ω)D-ωU]xk+ωb E=(D+ ωL)-1[(1-ω)D-ωU] f= ω(D+ ωL)-1b 13
  • 14. The Conjugate Gradient Method d 0 = r0 = b − Ax0 T ri ri αi = T d i Ad i xi +1 = xi + α i Ad i • Converges if A is a symmetric positive definite matrix • Convergence is faster T i +1 i +1 T i i r r βi +1 = r r d i +1 = ri +1 + βi +1d i 14
  • 15. Convergence of Iterative Methods ˆ Define the solution vector as x k Define an error vector as e ˆ x =e +x k k Substitute this into x k +1 = Ex + f k ˆ ˆ ˆ e k +1 + x = E (e k + x) + f = Ex + f + Ee k e k +1 = Ee k = EEe k −1 = EEEe k − 2 = E ( k +1) e 0 15
  • 16. Convergence of Iterative Methods iteration e k +1 = E ( k +1) 0 e ≤ E ( k +1) e 0 power The iterative method will converge for any initial iteration vector if the following condition is satisfied Convergence condition Lim e k +1 → 0 if k →∞ Lim E ( k +1) → 0 k →∞ 16
  • 17. Norm of a vector A vector norm should satisfy these conditions x ≥ 0 for every nonzero vector x x = 0 iff x is a zero vector αx = α x for scalar α x+ y ≤ x + y Vector norms can be defined in different forms as long as the norm definition satisfies these conditions 17
  • 18. Commonly used vector norms Sum norm or ℓ1 norm x 1 = x1 + x2 +  + xn Euclidean norm or ℓ2 norm 2 2 x 2 = x12 + x2 +  + xn Maximum norm or ℓ∞ norm x ∞ = max i xi 18
  • 19. Norm of a matrix A matrix norm should satisfy these conditions A ≥0 A = 0 iff A is a zero matrix for scalar α αA = α A A+ B ≤ A + B Important identitiy Ax ≤ A x x is a vector 19
  • 20. Commonly used matrix norms Maximum column-sum norm or ℓ1 norm m A 1 = max ∑ aij 1≤ j ≤ n i =1 Spectral norm or ℓ2 norm A 2 = maximum eigenvalue of AT A Maximum row-sum norm or ℓ∞ norm n A ∞ = max ∑ aij 1≤i ≤ m j =1 20
  • 21. Example  Compute the ℓ1 and ℓ∞ norms of the matrix 3 9 5 7 2 4    6 8 1    16 19 17 = A ∞ 13 15 10 = A1 21
  • 22. Convergence condition lim e k →∞ k +1 → 0 if lim E k →∞ ( k +1) →0 Express E in terms of modal matrix P and Λ Λ:Diagonal matrix with eigenvalues of E on the diagonal E = PΛP −1 E ( k +1) = PΛP −1 PΛP −1  PΛP −1 E ( k +1) = PΛ( k +1) P −1 k λ1 +1    λk +1 2  Λk +1 =       λk +1   n   lim E ( k +1) → 0 ⇒ lim PΛ( k +1) P −1 → 0 ⇒ lim Λ( k +1) → 0 k →∞ k →∞ k →∞ ⇒ lim λki +1 → 0 ⇒ λ i < 1 for i = 1,2 ,...,n k →∞ 22
  • 23. Sufficient condition for convergence If the magnitude of all eigenvalues of iteration matrix E is less than 1 than the iteration is convergent It is easier to compute the norm of a matrix than to compute its eigenvalues Ex = λx Ex = λ x   ⇒ λ x ≤ E x ⇒ λ ≤ E Ex ≤ E x   E < 1 is a sufficient condition for convergence 23
  • 24. Convergence of Jacobi iteration E=-D-1(L+U)   0   − a21  a22 E=      an1 − a  nn a12 − a11 0   a23 − a22         ann −1 − ann a1n  − a11   a2 n  − a22    an −1n  −  an −1n −1   0   24
  • 25. Convergence of Jacobi iteration Evaluate the infinity(maximum row sum) norm of E E n ∞ <1⇒ ∑ j =1 i≠ j aij < 1 for i = 1,2,..., n aii n ⇒ aii > ∑ aij j =1 i≠ j Diagonally dominant matrix If A is a diagonally dominant matrix, then Jacobi iteration converges for any initial vector 25
  • 26. Stopping Criteria  Ax=b  At any iteration k, the residual term is rk=b-Axk  Check the norm of the residual term ||b-Axk||  If it is less than a threshold value stop 26
  • 27. Example 1 (Jacobi Iteration)  4 − 1 1  x1   7   4 − 8 1  x  = − 21   2    − 2 1 5  x3   15       0  x 0 = 0    0    b − Ax 0 2 = 26.7395 Diagonally dominant matrix 0 0 7 + x2 − x3 7 x = = = 1.75 4 4 0 21 + 4 x10 + x3 21 x1 = = = 2.625 2 8 8 0 15 + 2 x10 − x2 15 1 x3 = = = 3.0 5 5 1 1 b − Ax1 = 10.0452 2 27
  • 28. Example 1 continued... 1 7 + x1 − x3 7 + 2.625 − 3 2 x = = = 1.65625 4 4 1 1 21 + 4 x1 + x3 21 + 4 × 1.75 + 3 2 x2 = = = 3.875 8 8 1 15 + 2 x1 − x1 15 + 2 × 1.75 − 2.625 2 2 = = 4.225 x3 = 5 5 2 1 7 + 3.875 − 4.225 = 1.6625 4 21 + 4 ×1.65625 + 4.225 3 x2 = = 3.98125 8 15 + 2 × 1.65625 − 3.875 3 x3 = = 2.8875 5 b − Ax 2 2 = 6.7413 x13 = b − Ax 2 2 = 1.9534 Matrix is diagonally dominant, Jacobi iterations are converging 28
  • 29. Example 2 − 2 1 5  x1   15   4 − 8 1  x  = − 21   2     4 − 1 1  x3   7       0  x 0 = 0    0    b − Ax 0 2 = 26.7395 The matrix is not diagonally dominant 0 0 − 15 + x2 + 5 x3 − 15 x = = = −7.5 2 2 0 21 + 4 x10 + x3 21 1 x2 = = = 2.625 8 8 1 0 x3 = 7 − 4 x10 + x2 = 7.0 1 1 b − Ax1 = 54.8546 2 29
  • 30. Example 2 continued... − 15 + 2.625 + 5 × 7 = 11.3125 2 21 − 4 × 7.5 + 7 x1 = = −0.25 2 8 1 x3 = 7 + 4 × 7.5 + 2.625 = 39.625 1 x1 = b − Ax 2 2 = 208.3761 The residual term is increasing at each iteration, so the iterations are diverging. Note that the matrix is not diagonally dominant 30
  • 31. Convergence of Gauss-Seidel iteration    GS iteration converges for any initial vector if A is a diagonally dominant matrix GS iteration converges for any initial vector if A is a symmetric and positive definite matrix Matrix A is positive definite if xTAx>0 for every nonzero x vector 31
  • 32. Positive Definite Matrices    A matrix is positive definite if all its eigenvalues are positive A symmetric diagonally dominant matrix with positive diagonal entries is positive definite If a matrix is positive definite   All the diagonal entries are positive The largest (in magnitude) element of the whole matrix must lie on the diagonal 32
  • 33. Positive Definitiness Check 20 12 25 12 15 2     25 2 5    Not positive definite Largest element is not on the diagonal 5 20 12 12 − 15 2    5 2 25   Not positive definite All diagonal entries are not positive 20 12 5  12 15 2     5 2 25   Positive definite Symmetric, diagonally dominant, all diagonal entries are positive 33
  • 34. Positive Definitiness Check 20 12 5  12 15 2     8 2 25   A decision can not be made just by investigating the matrix. The matrix is diagonally dominant and all diagonal entries are positive but it is not symmetric. To decide, check if all the eigenvalues are positive 34
  • 35. Example (Gauss-Seidel Iteration)  4 − 1 1  x1   7   4 − 8 1  x  = − 21   2    − 2 1 5  x3   15       0  x 0 = 0    0    b − Ax 0 2 = 26.7395 Diagonally dominant matrix 0 0 7 7 + x2 − x3 = = 1.75 x = 4 4 1 0 21 + 4 x1 + x3 21 + 4 × 1.75 1 = = 3 .5 x2 = 8 8 1 15 + 2 x1 − x1 15 + 2 × 1.75 − 3.5 1 2 = = 3 .0 x3 = 5 5 1 1 b − Ax1 = 3.0414 2 b − Ax1 = 10.0452 2 Jacobi iteration 35
  • 36. Example 1 continued... 1 7 + 3.5 − 3 7 + x1 − x3 2 x = = = 1.875 4 4 1 21 + 4 x12 + x3 21 + 4 × 1.875 + 3 2 x2 = = = 3.9375 8 8 2 15 + 2 x12 − x2 15 + 2 × 1.875 − 3.9375 2 = = 2.9625 x3 = 5 5 2 1 b − Ax 2 b − Ax 2 2 2 = 0.4765 = 6.7413 Jacobi iteration When both Jacobi and Gauss-Seidel iterations converge, Gauss-Seidel converges faster 36
  • 37. Convergence of SOR method    If 0<ω<2, SOR method converges for any initial vector if A matrix is symmetric and positive definite If ω>2, SOR method diverges If 0<ω<1, SOR method converges but the convergence rate is slower (deceleration) than the Gauss-Seidel method. 37
  • 38. Operation count     The operation count for Gaussian Elimination or LU Decomposition was 0 (n3), order of n3. For iterative methods, the number of scalar multiplications is 0 (n2) at each iteration. If the total number of iterations required for convergence is much less than n, then iterative methods are more efficient than direct methods. Also iterative methods are well suited for sparse matrices 38