Engineering Computation Part 3 E. T. S. I. Caminos, Canales y Puertos

Learning Objectives for Lecture 1. Motivate Study of Systems of Equations and particularly Systems of Linear Equations 2. Review steps of Gaussian Elimination 3. Examine how roundoff error can enter and be magnified in Gaussian Elimination 4. Introduce Pivoting and Scaling as defenses against roundoff. 5. Consider what an engineer can do to generate well formulated problems. E. T. S. I. Caminos, Canales y Puertos

Systems of Equations In Part 2 we have tried to determine the value x, satisfying f(x)=0. In this part we try to obtain the values x1,x2, xn, satisfying the system of equations: These systems can be linear or nonlinear, but in this part we deal with linear systems: E. T. S. I. Caminos, Canales y Puertos

Systems of Equations where a and b are constant coefficients, and n is the number of equations. Many of the engineering fundamental equations are based on conservation laws. In mathematical terms, these principles lead to balance or continuity equations relating the system behavior with respect to the amount of the magnitude being modelled and the extrenal stimuli acting on the system. E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Column 3 Row 2 Matrices are rectangular sets of elements represented by a single symbol. If the set if horizontal it is called row, and if it is vertical, it is called column. Column 3 Row 2 Column vector Row vector E. T. S. I. Caminos, Canales y Puertos

Systems of Equations There are some special types of matrices: Identity matrix Symmetric matrix Diagonal matrix Upper triangular matrix E. T. S. I. Caminos, Canales y Puertos

Lower triangular matrix Systems of Equations Lower triangular matrix Banded matrix Half band width All elements are null with the exception of thoise in a band centered around the main diagonal. This matrix has a band width of 3 and has the name of tridiagonal. E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Linear Algebraic Equations a11x1 + a12x2 + a13x3 + … + a1nxn = b1 a21x1 + a22x2 + a23x3 + … + a2nxn = b2 ….. an1x1 + an2x2 + an3x3 + … + anxn = bn where all aij's and bi's are constants. In matrix form: n x n n x 1 n x 1 or simply [A]{x} = {b} E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Matrix product: Matrix representation of a system Resulting dimensions E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Graphic Solution: Systems of equations are hyperplanes (straight lines, planes, etc.). The solution of a system is the intersection of these hyperplanes. Compatible and determined system. Vectors are linearly independent. Unique solution. Determinant of A is non-null. E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Incompatible system, Linearly dependent vectors. Null determinant of A. There is no solution. Compatible but undetermined system. Linearly dependent vectors. Null determinant of A. There exists an infinite number of solutions. E. T. S. I. Caminos, Canales y Puertos

Systems of Equations Compatible and determined system. Linearly independent vectors. Nonnull determinant of A, but close to zero. There exists a solution but it is difficult to find precisely. It is an ill conditioned system leading to numerical errors. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination Naive Gauss elimination method: The Gauss’ method has two phases: Forward elimination and backsustitution. In the first, the system is reduced to an upper triangular system: First, the unknown x1 is eliminated. To this end, the first row is multiplied by -a21/a11 and added to the second row. The same is done with all other succesive rows (n-1 times) until only the first equation contains the first unknown x1. Pivot equation substract pivot E. T. S. I. Caminos, Canales y Puertos

Gauss elimination This operation is repeated with all variables xi, until an upper triangular matrix is obtained. Next, the system is solved by backsustitution. The number of operations (FLOPS) used in the Gauss method is: Pass 1 Pass 2 E. T. S. I. Caminos, Canales y Puertos

Gauss elimination 1. Forward Elimination (Row Manipulation): a. Form augmented matrix [A|b]: b. By elementary row manipulations, reduce [A|b] to [U|b'] where U is an upper triangular matrix: DO i = 1 to n-1 DO k = i+1 to n Row(k) = Row(k) - (aki/aii)*Row(i) ENDDO E. T. S. I. Caminos, Canales y Puertos

Gauss elimination 2. Back Substitution Solve the upper triangular system [U]{x} = {b´} xn = b'n / unn DO i = n-1 to 1 by (-1) END E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Consider the system of equations To 2 significant figures, the exact solution is: We will use 2 decimal digit arithmetic with rounding. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Start with the augmented matrix: Multiply the first row by –1/50 and add to second row. Multiply the first row by –2/50 and add to third row: Multiply the second row by –6/40 E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Now backsolve: (vs. 0.091, et = 2.2%) (vs. 0.041, et = 2.5%) (vs. 0.016, et = 0%) E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Consider an alternative solution interchanging rows: After forward elimination, we obtain: Now backsolve: x3 = 0.095 (vs. 0.091, et = 4.4%) x2 = 0.020 (vs. 0.041, et = 50%) x1 = 0.000 (vs. 0.016, et = 100%) Apparently, the order of the equations matters! E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) WHAT HAPPENED? When we used 50 x1 + 1 x2 + 2 x3 = 1 to solve for x1, there was little change in other equations. When we used 2 x1 + 6 x2 + 30 x3 = 3 to solve for x1 it made BIG changes in the other equations. Some coefficients for other equations were lost! The second equation has little to do with x1. It has mainly to do with x3. As a result we obtained LARGE numbers in the table, significant roundoff error occurred and information was lost. Things didn't go well! If scaling factors | aji / aii | are  1 then the effect of roundoff errors is diminished. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Effect of diagonal dominance: As a first approximation roots are: xi  bi / aii Consider the previous examples: E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (example) Goals: 1. Best accuracy (i.e. minimize error) 2. Parsimony (i.e. minimize effort) Possible Problems: A. Zero on diagonal term  ÷ by zero. B. Many floating point operations (flops) cause numerical precision problems and propagation of errors. C. System may be ill-conditioned: det[A]  0. D. No solution or an infinite # of solutions: det[A] = 0. Possible Remedies: A. Carry more significant figures (double precision). B. Pivot when the diagonal is close to zero. C. Scale to reduce round-off error. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (pivoting) A. Row pivoting (Partial Pivoting) - In any good routine, at each step i, find maxk | aki | for k = i, i+1, i+2, ..., n Move corresponding row to pivot position. (i) Avoids zero aii (ii) Keeps numbers small & minimizes round-off, (iii) Uses an equation with large | aki | to find xi Maintains diagonal dominance. Row pivoting does not affect the order of the variables. Included in any good Gaussian Elimination routine. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (pivoting) B. Column pivoting - Reorder remaining variables xj for j = i, . . . ,n so get largest | aji | Column pivoting changes the order of the unknowns, xi, and thus leads to complexity in the algorithm. Not usually done. C. Complete or Full pivoting Performing both row pivoting and column pivoting. (If [A] is symmetric, needed to preserve symmetry.) E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (pivoting) How to fool pivoting: Multiply the third equation by 100 and then performing pivoting will yield: Forward elimination then yields (2-digit arithmetic): Backsolution yields: x3 = 0.095 (vs. 0.091, et = 4.4%) x2 = 0.020 (vs. 0.041, et = 50.0%) x1 = 0.000 (vs. 0.016, et = 100%) The order of the rows is still poor!! E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (scaling) A. Express all equations (and variables) in comparable units so all elements of [A] are about the same size. B. If that fails, and maxj |aij| varies widely across the rows, replace each row i by: aij  This makes the largest coefficient |aij| of each equation equal to 1 and the largest element of [A] equal to 1 or -1 NOTE: Routines generally do not scale automatically; scaling can cause round-off error too! SOLUTIONS • Don't actually scale, but use hypothetical scaling factors to determine what pivoting is necessary. • Scale only by powers of 2: no roundoff or division required. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (scaling) How to fool scaling: A poor choice of units can undermine the value of scaling. Begin with our original example: If the units of x1 were expressed in µg instead of mg the matrix might read: Scaling then yields: Which equation is used to determine x1 ? Why bother to scale ? E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) In numerical scientific calculations, the number of multiplies & divides often determines CPU time. (This represents the numerical effort!) One floating point multiply or divide (plus any associated adds or subtracts) is called a FLOP. (The adds/subtracts use little time compared to the multiplies/divides.) FLOP = FLoating point OPeration. Examples: a * x + b a / x – b E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Useful identities in counting FLOPS: O(mn) means that there are terms of order mn and lower. E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Simple Example of Operation Counting: DO i = 1 to n Y(i) = X(i)/i – 1 ENDDO X(i) and Y(i) are arrays whose values change when i changes. In each iteration X(i)/i – 1 represents one FLOP because it requires one division (& one subtraction). The DO loop extends over i from 1 to n iterations: E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Another Example of Operation Counting: DO i = 1 to n Y(i) = X(i) X(i) + 1 DO j = i to n Z(j) = [ Y(j) / X(i) ] Y(j) + X(i) ENDDO With nested loops, always start from the innermost loop. [Y(j)/X(i)] * Y(j) + X(i) represents 2 FLOPS E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) For the outer i-loop: X(i) • X(i) + 1 represents 1 FLOP = 3n +2n2 - n2 - n = n2 + 2n = n2 + O(n) E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Forward Elimination: DO k = 1 to n–1 DO i = k+1 to n r = A(i,k)/A(k,k) DO j = k+1 to n A(i,j)=A(i,j) – r*A(k,j) ENDDO B(i) = B(i) – r*B(k) E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Back Substitution: X(n) = B(n)/A(n,n) DO i = n–1 to 1 by –1 SUM = 0 DO j = i+1 to n SUM = SUM + A(i,j)*X(j) ENDDO X(i) = [B(i) – SUM]/A(i,i) E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Forward Elimination Inner loop: Second loop: = (n2 + 2n) – 2(n + 1)k + k2 E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Forward Elimination (cont'd) Outer loop = E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Back Substitution Inner Loop: Outer Loop: E. T. S. I. Caminos, Canales y Puertos

Gauss elimination (operation counting) Total flops = Forward Elimination + Back Substitution = n3/3 + O (n2) + n2/2 + O (n)  n3/3 + O (n2) To convert (A,b) to (U,b') requires n3/3, plus terms of order n2 and smaller, flops. To back solve requires: 1 + 2 + 3 + 4 + . . . + n = n (n+1) / 2 flops; Grand Total: the entire effort requires n3/3 + O(n2) flops altogether. E. T. S. I. Caminos, Canales y Puertos

Gauss-Jordan Elimination Diagonalization by both forward and backward elimination in each column. Perform elimination both backwards and forwards until: Operation count for Gauss-Jordan is: (slower than Gauss elimination) E. T. S. I. Caminos, Canales y Puertos

Gauss-Jordan Elimination Example (two-digit arithmetic): x1 = 0.015 (vs. 0.016, et = 6.3%) x2 = 0.041 (vs. 0.041, et = 0%) x3 = 0.093 (vs. 0.091, et = 2.2%) E. T. S. I. Caminos, Canales y Puertos

Gauss-Jordan Matrix Inversion The solution of: [A]{x} = {b} is: {x} = [A]-1{b} where [A]-1 is the inverse matrix of [A] Consider: [A] [A]-1 = [ I ] 1) Create the augmented matrix: [ A | I ] 2) Apply Gauss-Jordan elimination: ==> [ I | A-1 ] E. T. S. I. Caminos, Canales y Puertos

Gauss-Jordan Matrix Inversion Gauss-Jordan Matrix Inversion (with 2 digit arithmetic): MATRIX INVERSE [A-1] E. T. S. I. Caminos, Canales y Puertos

Gauss-Jordan Matrix Inversion CHECK: [ A ] [ A ]-1 = [ I ] [ A ]-1 { b } = { x } Gaussian Elimination E. T. S. I. Caminos, Canales y Puertos

LU decomposition LU decomposition - The LU decomposition is a method that uses the elimination techniques to transform the matrix A in a product of triangular matrices. This is specially useful to solve systems with different vectors b, because the same decomposition of matrix A can be used to evaluate in an efficient form, by forward and backward sustitution, all cases. E. T. S. I. Caminos, Canales y Puertos

LU decomposition Initial system Decomposition Transformed system 1 Substitution Transformed system 2 Backward sustitution Forward sustitution E. T. S. I. Caminos, Canales y Puertos

LU decomposition LU decomposition is very much related to Gauss method, because the upper triangular matrix is also looked for in the LU decomposition. Thus, only the lower triangular matrix is needed. Surprisingly, during the Gauss elimination procedure, this matrix L is obtained, but one is not aware of this fact. The factors we use to get zeroes below the main diagonal are the elements of this matrix L. Substract E. T. S. I. Caminos, Canales y Puertos

LU decomposition resto resto E. T. S. I. Caminos, Canales y Puertos

LU decomposition (Complexity) Basic Approach Consider [A]{x} = {b} a) Gauss-type "decomposition" of [A] into [L][U] n3/3 flops [A]{x} = {b} becomes [L] [U]{x} = {b}; let [U]{x}  {d} b) First solve [L] {d} = {b} for {d} by forward subst. n2/2 flops c) Then solve [U]{x} = {d} for {x} by back substitution n2/2 flops E. T. S. I. Caminos, Canales y Puertos

LU Decompostion: notation [A] = [L] + [U0] [A] = [L0] + [U] [A] = [L0] + [U0] + [D] [A] = [L1] [U] [A] = [L] [U1] E. T. S. I. Caminos, Canales y Puertos

LU decomposition LU Decomposition Variations Doolittle [L1][U] General [A] Crout [L][U1] General [A] Cholesky [L][L] T Pos. Def. Symmetric [A] Cholesky works only for Positive Definite Symmetric matrices Doolittle versus Crout: • Doolittle just stores Gaussian elimination factors where Crout uses a different series of calculations (see C&C 10.1.4). • Both decompose [A] into [L] and [U] in n3/3 FLOPS • Different location of diagonal of 1's • Crout uses each element of [A] only once so the same array can be used for [A] and [L\U] saving computer memory! E. T. S. I. Caminos, Canales y Puertos

Definition of a matrix inverse: LU decomposition Matrix Inversion Definition of a matrix inverse: [A] [A]-1 = [ I ] ==> [A] {x} = {b} [A]-1 {b} = {x} First Rule: Don’t do it. (numerically unstable calculation) E. T. S. I. Caminos, Canales y Puertos

LU decomposition Matrix Inversion If you really must -- 1) Gaussian elimination: [A | I ] –> [U | B'] ==> A-1 2) Gauss-Jordan: [A | I ] ==> [I | A-1 ] Inversion will take n3 + O(n2) flops if one is careful about where zeros are (taking advantage of the sparseness of the matrix) Naive applications (without optimization) take 4n3/3 + O(n2) flops. For example, LU decomposition requires n3/3 + O(n2) flops. Back solving twice with n unit vectors ei: 2 n (n2/2) = n3 flops. Altogether: n3/3 + n3 = 4n3/3 + O(n2) flops E. T. S. I. Caminos, Canales y Puertos

FLOP Counts for Linear Algebraic Equations Summary FLOP Counts for Linear Algebraic Equations, [A]{x} = {b} Gaussian Elimination (1 r.h.s) n3/3 + O (n2) Gauss-Jordan (1 r.h.s) n3/2 + O (n2) LU decomposition n3/3 + O (n2) Each extra LU right-hand-side n2 Cholesky decomposition (symmetric A) n3/6 + O (n2) Inversion (naive Gauss-Jordan) 4n3/3 +O (n2) Inversion (optimal Gauss-Jordan) n3 + O (n2) Solution by Cramer's Rule n! 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

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 x2 b2/a22 b1/a11 b1/a12 a11x1+ a12x2 = b1 a21x1+ a22x2 = b2 b2/a21 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 change in 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 "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 Matrix Norms 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

E. T. S. I. Caminos, Canales y Puertos Linfield & Penny, 1999

E. T. S. I. Caminos, Canales y Puertos Linfield & Penny, 1999

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 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