Lecture Notes Course ÃMA 190 Numerical Mathematics, First ...

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We now determine c such that 2 + 3c = 0, i.e. c = −2/3, and hence arrive atthe system3x + y = 1, −5/3 · y = 10/3 . (3.6)This last system is of triangular form and we find easily y = −2 and subsequentlyx = 1 This procedure is applied to general systems of equations to transformthem into systems of a special structure whose solution set is easily determined.This scheme which is called Gaussian elimination with pivoting is described insection 3.33.2 Trapezoidal systemsDefinition 3.2.1 Let A be a rectangular matrix with n rows and m columns,x and b column vectors with m and n elements respectively. ThenAx = b,specifies a linear system of equations with n equations and m unknowns. b isthe right hand side, A the table of coefficients and x is a solution vector to bedetermined.The system may be inconsistent, have a unique solution or have infinitely manysolution vectors.If n = m, the matrix A is called square. The element in row r and column k isdenoteda r,k .The matrix A is called trapezoidal or in echelon form if it is such that: There isan integer k with the properties (1) and (2) below:(1)a i,i ≠ 0, a i,j = 0, j < i, i = 1, . . . , k.(2)If k < n thena i,j = 0, i > k.Important special case: If m = n = k then A is said to be triangular.Example 3.2.2⎛1 2 3⎞ ⎛A = ⎝ 0 4 5 ⎠ B = ⎝0 0 61 2 3 70 4 5 80 0 6 9⎞⎛⎠ C = ⎝1 2 30 4 50 0 0All the three matrices above are trapezoidal. Matrix A is even triangular.Consider now the three linear systems of equationsAx = b, By = b, Cz = b.⎞⎠ .27
Then we easily verify: The system Ax = b has a unique solution for all righthand sides b. The system By = b has infinitely many solutions since y 4 maybe chosen arbitrarily and then the remaining elements in the solution y may beexpressed uniquely in terms of y 4 . The system Cz = b is solvable only if b 3 = 0.In this case z 3 may be chosen arbitrarily and the remaining components of thesolution vector may be expressed uniquely in terms of z 3 .Definition 3.2.3 The following two classes of square matrices are useful: Thesquare matrix A is termed symmetric ifA is called tri-diagonal ifa i,j = a j,i .a i,j = 0, |i − j| > 1 .Provided that the table of coefficients is of trapezoidal form, it is possible todetermine all solutions of the systemAx = b,by means of a finite number of arithmetic operations.3.3 Reduction of a general system of equationsto trapezoidal formIf A is of trapezoidal form, then the system Ax = b is such that the coefficienta i,j in front of the unknown x j in equation number i is 0 if i > j. The Gaussianelimination scheme may be used to reduce a general system of equations on asystem of the trapezoidal form which has the same solution set as the originalsystem. To secure stability, pivoting, i.e. reordering of the equations, is performedaccording to certain rules to be described below. For certain classes ofmatrices pivoting is not required. Also, if the elimination is carried out exactly,as e.g. in theoretical analysis, pivoting is only carried out to avoid division by0.We describe now the Gaussian elimination for a general system Ax = b, whereA has n rows and m columns. The main idea is that by performing eliminationstep number k one eliminates the coefficient in front of the unknown x k in equationsi > k for k = 1, . . . , min(m, n). It is essential that the coefficient in frontof x k in equation k remains different from 0. The algorithm may be describedas follows: (In order to simplify the notation we denote by a i,j the contents ofthe place in row i and column j, which could be thought of as a space in thecomputer.)Step 1: Elimination of coefficients in front of x 1 :Substep 1a: Exceptional case: If all element in column 1 are 0, find another columnwhich has a nonzero element and interchange it with the first column. (Observethat this interchange means a reordering of the unknowns, which should28
Page 1: Lecture NotesCourse ÅMA 190Numeric
Page 6 and 7: 13.8.4 Inverse linear interpolation
Page 8 and 9: AbstractNumerical analysts study th
Page 10 and 11: 1.1.1 Two standard inequalitiesLet
Page 12 and 13: Example 1.1.3 Examples with s = 3:N
Page 14 and 15: and the relative condition number b
Page 16 and 17: This general example may be special
Page 18 and 19: Theorem 1.5.2 Let A be a square mat
Page 20 and 21: 1.6.2 Arithmetic sequencen(n − 1)
Page 22 and 23: 1.6.7 Taylor’s formulaWe derive T
Page 24 and 25: )c)becauselimx→0x 2e x − 1 −
Page 26 and 27: 1.7.7 Locating rootsPutShow that th
Page 28 and 29: We next construct the general solut
Page 30 and 31: Definition 2.2.1 We distinguish bet
Page 32 and 33: Example 2.3.5 Consider a computer w
Page 36 and 37: e recorded) If there is no such col
Page 38 and 39: the left hand side and the right ha
Page 40 and 41: Chapter 4Iterative methods for Ax =
Page 42 and 43: or, in matrix form with the coeffic
Page 44 and 45: Chapter 5Least squares fit5.1 Norma
Page 46 and 47: c 1,4 is the product of the first a
Page 48 and 49: PutandL i (t) =P (t) =5∏(t − t
Page 50 and 51: 6.2 On the choice of nodesIf t i in
Page 52 and 53: k arguments:f[t i1 , . . . , t ik ]
Page 54 and 55: Q(x) = 4.6 − 0.8(x + 2) + 0.6x(x
Page 56 and 57: a relative error bound if x ≠ 0.
Page 58 and 59: 7.2.2 MultiplicationPutwheres = x 1
Page 60 and 61: We now assume that f has a continuo
Page 62 and 63: functional values. Let s ∗ be an
Page 64 and 65: 7.5.2 Example: additionLetWe then h
Page 66 and 67: Chapter 8Numerical treatment ofnonl
Page 68 and 69: 8.3 Determination of approximate ro
Page 70 and 71: Proof:The case n = 0 is the stateme
Page 72 and 73: 8.6 The fixed-point methodTo use th
Page 74 and 75: Chapter 9On approximation offunctio
Page 76 and 77: 9.1.3 The square rootTo find √ a,
Page 78 and 79: which is achieved forTo gett i = 1/
Page 80 and 81: whereF (h, w) = h∞∑n=−∞e iw
Page 82 and 83: Proof PutThen we haveandM n = supr>
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10.3 Numerical treatment of rapidly
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n a n−1 s n1 1.00000 1.000002 −
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phenomenon we needDefinition 10.4.1
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a r and b r are given bya r = exp(
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11.2 Constructing the trapezoidal r
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In the case of (11.2) it is suffici
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should give exact results for all l
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h=0.2 :f(0) + f(1.6)T (0.2) = 0.2
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h=0.5 :T (0.5) = 0.5 · f(0) + f(0.
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A system of ordinary differential e
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12.3.1 Euler’s methodx r+1 = x r
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It is directly verified that the so
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1.00 1.246834041.00 1.246818301.00
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13.1.3 Stirling’s formula13.1.4 B
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Power expansions for some elementar
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TranspositionThenA ∈ R n×m , B =
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13.5 Least squares solution of line
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Let f be a function which may be di
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13.9 Systems of nonlinear equations
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may be writtenx(t) = x h (t) + x p
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Chapter 14Laboratory exercises andc
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Start with x 0 = 1b) Use the bisect
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14.7 Case studies14.8 Evaluation of
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n piv,sing nonpiv,sing piv,dbl nonp
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We put c = 100 + u and solve the pr
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and hence we may consider minimisin
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x f(x)−1.0 0.3679−0.5 0.6065 .0
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14.12 Numerical integration problem
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14.12.5 Integrand with integrable s
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Lecture Notes Course ÃMA 190 Numerical Mathematics, First ...