Engineering Computation

EngineeringComputation Lecture 7 E. T. S. I. Caminos, Canales y Puertos

Errors in Solutions to Systems of Linear Equations • System of Equations • Errors in Solutions to Systems of Linear Equations • Objective:Solve [A]{x} = {b} • Problem: • Round-off errors may accumulate and even be exaggerated by the solution procedure. Errors are often exaggerated if the system is ill-conditioned • Possible remedies to minimize this effect: • 1. Partial or complete pivoting • 2. Work in double precision • 3. Transform the problem into an equivalent system of linear equations by scaling or equilibrating E. T. S. I. Caminos, Canales y Puertos

Errors in Solutions to Systems of Linear Equations • Ill-conditioning • A system of equations is singular if det|A| = 0 • If a system of equations is nearly singular it is ill-conditioned. • Systems which are ill-conditioned are extremely sensitive to small changes in coefficients of [A] and {b}. These systems are inherently sensitive to round-off errors. • Question: • Can we develop a means for detecting these situations? E. T. S. I. Caminos, Canales y Puertos

a11x1+ a12x2 = b1 x2 b2/a22 b2/a21 b1/a12 x1 b1/a11 a21x1+ a22x2 = b2 Errors in Solutions to Systems of Linear Equations Ill-conditioning of [A]{x} = {b}: Consider the graphical interpretation for a 2-equation system: We can plot the two linear equations on a graph of x1 vs. x2. E. T. S. I. Caminos, Canales y Puertos

Errors in Solutions to Systems of Linear Equations Ill-conditioning of [A]{x} = {b}: Consider the graphical interpretation for a 2-equation system: We can plot the two linear equations on a graph of x1 vs. x2. x1 x1 x2 x2 Uncertainty in x2 Uncertainty in x2 Well-conditioned Ill-conditioned E. T. S. I. Caminos, Canales y Puertos

Errors in Solutions to Systems of Linear Equations Ways to detect ill-conditioning: 1. Calculate {x}, make small change in [A] or {b} and determine the change in the solution {x}. 2. After forward elimination, examine diagonal of upper triangular matrix. If aii << ajj, i.e. there is a relatively small value on diagonal, then this may indicate ill-conditioning. 3. Compare {x}single with {x}double 4. Estimate the "condition number" for A. Substituting the calculated {x} into [A]{x} and checking this against {b} will not always work!!! E. T. S. I. Caminos, Canales y Puertos

Errors in Solutions to Systems of Linear Equations • Ways to detect ill-conditioning: • If det|A| = 0 the matrix is singular • ==> the determinant may be an indicator of conditioning • If det|A| is near zero is the matrix ill-conditioned? • Consider: • After scaling: • ==> det|A| will provide an estimate of conditioning if it is normalized by the "magnitude" of the matrix. E. T. S. I. Caminos, Canales y Puertos

Norms Norms and the Condition Number We need a quantitative measure of ill-conditioning. This measure will then directly reflect the possible magnitude of round off effects. To do this we need to understand norms: Norm:Scalar measure of the magnitude of a matrix or vector ("how big" a vector is). Not to be confused with the dimension of a matrix. E. T. S. I. Caminos, Canales y Puertos

Vector Norms Vector Norms: Scalar measure of the magnitude of a vector Here are some vector norms for n x 1 vectors {x} with typical elements xi. Each is in the general form of a p norm defined by the general relationship: 1. Sum of the magnitudes: 2. Magnitude of largest element: (infinity norm) 3. Length or Euclidean norm: E. T. S. I. Caminos, Canales y Puertos

Norms • Vector Norms • Required Properties of vector norm: • 1. ||x||  0 and ||x|| = 0 if and only if [x]=0 • 2 ||kx|| = k ||x|| where k is any positive scalar • 3. ||x+y||  ||x|| + ||y|| Triangle Inequality • For the Euclidean vector norm we also have • 4. ||x•y||  ||x|| ||y|| • because the dot product or inner product property satisfies: • ||xy|| = ||x||•||y|| |cos()|  ||x|| • ||y||. E. T. S. I. Caminos, Canales y Puertos

Matrix Norms Matrix Norms: Scalar measure of the magnitude of a matrix. Matrix norms corresponding to vector norms above are defined by the general relationship: 1. Largest column sum: (column sum norm) 2. Largest row sum: (row sum norm) (infinity norm) E. T. S. I. Caminos, Canales y Puertos

Matrix norms • 3. Spectral norm: ||A|| 2 = (µmax)1/2 • where µmax is the largest eigenvalue of [A]T[A] • If [A] is symmetric, (µmax)1/2 = max , is the largest eigenvalue of [A]. • (Note: this is not the same as the Euclidean or Frobenius norm, seldom used: E. T. S. I. Caminos, Canales y Puertos

Matrix norms • MatrixNorms • For matrix norms to be useful we require that • 0. || Ax ||  || A || ||x || • General properties of any matrix norm: • 1. || A ||  0 and || A || = 0 iff [A] = 0 • 2. || k A || = k || A || where k is any positive scalar • 3. || A + B ||  || A || + || B || "Triangle Inequality" • 4. || A B ||  || A || || B || • Why are norms important? • Norms permit us to express the accuracy of the solution {x} in terms of ||x|| • Norms allow us to bound the magnitude of the product [ A ] {x} and the associated errors. E. T. S. I. Caminos, Canales y Puertos

Error Analysis • Forward and backward error analysis can estimate the effect of truncation and roundoff errors on the precision of a result. The two approaches are alternative views: • Forward (a priori) error analysis tries to trace the accumulation of error through each process of the algorithm, comparing the calculated and exact values at every stage. • Backward (a posteriori) error analysis views the final solution as the exact solution to a perturbed problem. One can consider how different the perturbed problem is from the original problem. • Here we use the condition number of a matrix [A] to specify the amount by which relative errors in [A] and/or {b} due to input, truncation, and rounding can be amplified by the linear system in the computation of {x}. E. T. S. I. Caminos, Canales y Puertos

Error Analysis • Backward Error Analysis of [A]{x} = {b} for errors in {b} • Suppose the coefficients {b} are not precisely represented. What might be the effect on the calculated value for {x + dx}? • Lemma: [A]{x} = {b} yields ||A|| ||x||  ||b|| or • Now an error in {b} yields a corresponding error in {x}: • [A ]{x + dx} = {b + db} • [A]{x} + [A]{ dx} = {b} + {db} • Subtracting [A]{x} = {b} yields: • [A]{dx} = {db} ––> {dx} = [A]-1{db} E. T. S. I. Caminos, Canales y Puertos

Error Analysis Backward Error Analysis of [A]{x} = {b} for errors in {b} Taking norms we have: And using the lemma: we then have : Define the condition number as k = cond [A]  ||A-1|| ||A||  1 If k 1 or k is small, the system is well-conditioned If k >> 1, system is ill conditioned. 1 = || I || = || A-1A ||  || A-1 || || A || = k = Cond(A) E. T. S. I. Caminos, Canales y Puertos

Error Analysis Backward Error Analysis of [A]{x} = {b} for errors in [A] If the coefficients in [A] are not precisely represented, what might be effect on the calculated value of {x+ dx}? [A + dA ]{x + dx} = {b} [A]{x} + [A]{ dx} + [dA]{x+dx} = {b} Subtracting [A]{x} = {b} yields: [A]{ dx} = – [dA]{x+dx} or {dx} = – [A]-1 [dA] {x+dx} Taking norms and multiplying by || A || / || A || yields : E. T. S. I. Caminos, Canales y Puertos

Error Analysis Estimate of Loss of Significance: Consider the possible impact of errors [dA] on the precision of {x}. • implies that if • Or, taking log of both sides: s > p - log10() • log10() is the loss in decimal precision; i.e., we start with p decimal figures and end-up with s decimal figures. • It is not always necessary to find [A]-1 to estimate k = cond[A]. Instead, use an estimate based upon iteration of inverse matrix using LU decomposition. E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods • Impetus for Iterative Schemes: • 1. May be more rapid if coefficient matrix is "sparse" • 2. May be more economical with respect to memory • 3. May also be applied to solve nonlinear systems • Disadvantages: • 1. May not converge or may converge slowly • 2. Not appropriate for all systems Error bounds apply to solutions obtained by direct and iterative methods because they address the specification of [dA] and {db}. E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods Basic Mechanics: Starting with: a11x1 + a12x2 + a13x3 + ... + a1nxn = b1 a21x1 + a22x2 + a23x3 + ... + a2nxn = b2 a31x1 + a32x2 + a33x3 + ... + a3nxn = b3 : : an1x1 + an2x2 + an3x3 + ... + annxn = bn Solve each equation for one variable: x1 = [b1 – (a12x2 + a13x3 + ... + a1nxn )} / a11 x2 = [b2 – (a21x1 + a23x3 + ... + a2nxn )} / a22 x3 = [b3 – (a31x1 + a32x2 + ... + a3nxn )} / a33 : xn = [bn – (an1x2 + an2x3 + ... + an,n-1xn-1 )} / ann E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods • Start with initial estimate of {x}0. • Substitute into the right-hand side of all the equations. • Generate new approximation {x}1. • This is a multivariate one-point iteration: {x}j+1 = {g({x}j)} • Repeat process until the maximum number of iterations is reached or until: || xj+1 – xj ||  d + e || xj+1 || E. T. S. I. Caminos, Canales y Puertos

Convergence • To solve [A]{x} = {b} • Separate [A] into: [A] = [Lo] + [D] + [Uo] • [D] = diagonal (aii) • [Lo] = lower triangular with 0's on diagonal • [Uo] = upper triangular with 0's on diagonal • Rewrite system: • [A]{x} = ( [Lo] + [D] + [Uo] ){x} = {b} • [D]{x} + ( [Lo] + [Uo] ){x} = {b} • Iterate: • [D]{x}j+1 = {b} – ( [Lo]+[Uo] ) {x}j • {x}j+1 = [D]-1{b} – [D]-1 ( [Lo]+[Uo] ) {x}j • Iterations converge if: • || [D]-1 ( [Lo] + [Uo] ) || < 1 • (sufficient if equations are diagonally dominant) E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods – the Jacobi Method E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods -- Gauss-Seidel In most cases using the newest values within the right-hand side equations will provide better estimates of the next value. If this is done, then we are using the Gauss-Seidel Method: ( [Lo]+[D] ){x}j+1 = {b} – [Uo] {x}j or explicitly: If this is done, then we are using the Gauss-Seidel Method E. T. S. I. Caminos, Canales y Puertos

Iterative Solution Methods -- Gauss-Seidel If either method is going to converge, Gauss-Seidel will converge faster than Jacobi. Why use Jacobi at all? Because you can separate the n-equations into n independent tasks, it is very well suited computers with parallel processors. E. T. S. I. Caminos, Canales y Puertos

Convergence of Iterative Solution Methods • Rewrite given system: [A]{x} = { [B] + [E] } {x} = {b} • where [B] is diagonal, or triangular so we can solve [B]{y} = {g} quickly. Thus, • [B] {x}j+1= {b}– [E] {x}j • which is effectively: {x}j+1 = [B]-1 ({b} – [E] {x}j ) • True solution {x}c satisfies: {x}c = [B]-1 ({b} – [E] {x}c) • Subtracting yields: {x}c – {x}j+1= – [B]-1 [E] [{x}c – {x}j] • So ||{x}c – {x}j+1 ||  ||[B]-1 [E]|| ||{x}c – {x}j || • Iterations converge linearly if || [B]-1 [E] || < 1=> || ([D] + [Lo])-1 [Uo] || < 1 For Gauss-Seidel • => || [D] -1 ([Lo] + [Uo]) || < 1 For Jacobi E. T. S. I. Caminos, Canales y Puertos

Convergence of Iterative Solution Methods • Iterative methods will not converge for all systems of equations, nor for all possible rearrangements. • If the system is diagonally dominant, • i.e., | aii | > | aij | where i  j then with all < 1.0, i.e., small slopes. E. T. S. I. Caminos, Canales y Puertos

Convergence of Iterative Solution Methods A sufficient condition for convergence exists: • Notes: • 1. If the above does not hold, still may converge. • 2. This looks similar to infinity norm of [A] E. T. S. I. Caminos, Canales y Puertos

Improving Rate of Convergence of G-S Iteration • Relaxation Schemes: • where 0.0 < l 2.0 • (Usually the value of l is close to 1) • Underrelaxation ( 0.0 < l< 1.0 ) • More weight is placed on the previous value. Often used to: • - make non-convergent system convergent or • - to expedite convergence by damping out oscillations. • Overrelaxation ( 1.0 < l 2.0 ) • More weight is placed on the new value. Assumes that the new value is heading in right direction, and hence pushes new value close to true solution. • The choice of l is highly problem-dependent and is empirical, so relaxation is usually only used for often repeated calculations of a particular class. E. T. S. I. Caminos, Canales y Puertos

Why Iterative Solutions? • We often need to solve [A]{x} = {b} where n = 1000's • • Description of a building or airframe, • • Finite-Difference approximations to PDE's. Most of A's elements will be zero; a finite-difference approximation to Laplace's equation will have five aij0 in each row of A. • Direct method (Gaussian elimination) • • Requires n3/3 flops (say n = 5000; n3/3 = 4 x 1010 flops) • • Fills in many of n2-5n zero elements of A • Iterative methods (Jacobi or Gauss-Seidel) • • Never store [A] (say n = 5000; [A] would need 4n2 = 100 Mb) • • Only need to compute [A-B] {x}; and to solve [B]{xt+1} = {b} E. T. S. I. Caminos, Canales y Puertos

Why Iterative Solutions? • • Effort: • Suppose [B] is diagonal, • solving [B] {v} = {b} n flops • Computing [A-B] x 4n flops • For m iterations 5mn flops • For n = m = 5000, 5mn = 1.25x108 • At worst O(n2). E. T. S. I. Caminos, Canales y Puertos

Engineering Computation