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

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Chapter 5Least squares fit5.1 Normal equationsDefinition 5.1.1 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.We discuss here the case when n > m. Then the system is, as a rule, inconsistent,i.e. one cannot find any vector x such that Ax = b exactly. Instead, one seeksa vector x which minimizes the square of the length of the deviation vector.Hence we want to minimize(Ax − b) T (Ax − b).This problem always has a solution, which satisfies the normal equations,A T Ax = A T b.This system may be solved using Gaussian elimination, and it can be verifiedthat pivoting is not required for stability. We describe next how to form theelements in the matrix C = A T A and vector f = A T b. Using the definitions ofmatrix by matrix product and matrix by vector product we find the formulas:c i,j =n∑a l,i a l,j f i =l=137n∑a l,i b l .l=1
Thus the element c i,j is the scalar product of columns number i and j of matrixA, while f i is the scalar product of column number i of matrix A and theoriginal right hand side b. Since the resulting system A T Ax = A T b, often calledthe normal equations, always is consistent, it may have a unique solution orinfinitely many solution vectors.Example 5.1.2 We consider the same example as in Chapter 2, namely thetask to approximate exp(t) with a polynomial of degree less than 4. Now we doleast squares fit with respect to the set of the five points −1, −0.5, 0, 0.5, 1. Thuswe get an over-determined linear system with 5 equations and 4 unknowns.The following output was obtainedgiven system1.000000 -1.000000 1.000000 -1.000000 .3679001.000000 -.500000 .250000 -.125000 .6065001.000000 .000000 .000000 .000000 1.0000001.000000 .500000 .250000 .125000 1.6487001.000000 1.000000 1.000000 1.000000 2.718300normal equations:5.000000 .000000 2.500000 .000000 6.341400.000000 2.500000 .000000 2.125000 2.8715002.500000 .000000 2.125000 .000000 3.650000.000000 2.125000 .000000 2.031250 2.480675triangular system5.000000 .000000 2.500000 .000000 6.341400.000000 2.500000 .000000 2.125000 2.871500.000000 .000000 .875000 .000000 .479300.000000 .000000 .000000 .225000 .039900solution vector.994394 .997867 .547771 .177333The coefficients c i,j of the left hand side and f i of the right hand side of thenormal equations are obtained by forming the scalar products of the columnsof the given (overdetermined) system. Thus the coefficients of the first normalequation are the scalar products of the first column and the other columns andthe right hand side. Thusc 1,1 is the scalar product of the first column with itself:c 1,1 = 1 + 1 + 1 + 1 + 1 = 5c 1,2 is the scalar products of the first and the second columns:c 1,2 = −1 − 0.5 + 0 + 0.5 + 1 = 0c 1,3 is the product of the first and the third columns:c 1,3 = 1 + 0.25 + 0 + 0.25 + 1 = 2.538
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 34 and 35: We now determine c such that 2 + 3c
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 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>
Page 84 and 85: 10.3 Numerical treatment of rapidly
Page 86 and 87: n a n−1 s n1 1.00000 1.000002 −
Page 88 and 89: phenomenon we needDefinition 10.4.1
Page 90 and 91: a r and b r are given bya r = exp(
Page 92 and 93: 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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