In other words the graph of the polynomial should pass through the points

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "In other words the graph of the polynomial should pass through the points"

Transcription

1 Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form of regression), industrial design, signal processing (digital-to-analog conversion) and in numerical analysis. It is one of tose important recurring concepts in applied matematics. In tis capter, we will immediately put interpolation to use to formulate ig-order quadrature and differentiation rules. 3.1 Polynomial interpolation Given N + 1 points x j R, 0 j N, and sample values y j = f(x j ) of a function at tese points, te polynomial interpolation problem consists in finding a polynomial p N (x) of degree N wic reproduces tose values: y j = p N (x j ), j = 0,..., N. In oter words te grap of te polynomial sould pass troug te points N (x j, yj). A degree-n polynomial can be written as p N (x) = n=0 a nx n for some coefficients a 0,..., a N. For interpolation, te number of degrees of freedom (N + 1 coefficients) in te polynomial matces te number of points were te function sould be fit. If te degree of te polynomial is strictly less tan N, we cannot in general pass it troug te points (x j, y j ). We can still try to pass a polynomial (e.g., a line) in te best approximate manner, but tis is a problem in approximation rater tan interpolation; we will return to it later in te capter on least-squares. 1

2 CHAPTER 3. INTERPOLATION Let us first see ow te interpolation problem can be solved numerically in a direct way. Use te expression of p N into te interpolating equations y j = p N (x j ): N n=0 a x n n j = y j, j = 0,..., N. In tese N + 1 equations indexed by j, te unknowns are te coefficients a 0,..., a N. We are in presence of a linear system V a = y, N Vjn a n = y j, n=0 wit V te so-called Vandermonde matrix, V jn = x n j, i.e., 1 x N 0 x 0 V = 1 x 1 x N x N x N N We can ten use a numerical software like Matlab to construct te vector of abscissas x j, te rigt-and-side of values y j, te V matrix, and numerically solve te system wit an instruction like a = V \ y (in Matlab). Tis gives us te coefficients of te desired polynomial. Te polynomial can now be plotted in between te grid points x j (on a finer grid), in order to display te interpolant. Historically, matematicians suc as Lagrange and Newton did not ave access to computers to display interpolants, so tey found explicit (and elegant) formulas for te coefficients of te interpolation polynomial. It not only simplified computations for tem, but also allowed tem to understand te error of polynomial interpolation, i.e., te difference f(x) p N (x). Let us spend a bit of time retracing teir steps. (Tey were concerned wit applications suc as fitting curves to celestial trajectories.) We ll define te interpolation error from te uniform (L ) norm of te difference f p N : f p N := max f(x) p N (x), were te maximum is taken over te interval [x 0, x N ]. x 2

3 3.1. POLYNOMIAL INTERPOLATION Call P N te space of real-valued degree-n polynomials: N P N = { a n x n : a n R}. n=0 Lagrange s solution to te problem of polynomial interpolation is based on te following construction. Lemma 1. (Lagrange elementary polynomials) Let {x j, j = 0,..., N} be a collection of disjoint numbers. For eac k = 0,..., N, tere exists a unique degree-n polynomial L k (x) suc tat { 1 if j = k; L k (x j ) = δ jk = 0 if j = k. Proof. Fix k. L k as roots at x j for j = k, so L k must be of te form 1 L k (x) = C (x x j ). j=k Evaluating tis expression at x = x k, we get 1 = C 1 (x k x j ) C =. j=k (x k x j ) j=k Hence te only possible expression for L k is L k (x) = j=k(x x j ). j=k (x k x j ) Tese elementary polynomials form a basis (in te sense of linear algebra) for expanding any polynomial interpolant p N. 1 Tat s because, if we fix j, we can divide L k (x) by (x x j ), j = k. We obtain L k (x) = (x x j )q(x) + r(x), were r(x) is a remainder of lower order tan x x j, i.e., a constant. Since L k (x j ) = 0 we must ave r(x) = 0. Hence (x x j ) must be a factor of L k (x). Te same is true of any (x x j ) for j = k. Tere are N suc factors, wic exausts te degree N. Te only remaining degree of freedom in L k is te multiplicative constant. 3

4 CHAPTER 3. INTERPOLATION Teorem 4. (Lagrange interpolation teorem) Let {x j, j = 0,..., N} be a collection of disjoint real numbers. Let {y j, j = 0,..., N} be a collection of real numbers. Ten tere exists a unique p N P N suc tat p N (x j ) = y j, j = 0,..., N. Its expression is N p N (x) = y k L k (x), (3.1) k=0 were L k (x) are te Lagrange elementary polynomials. Proof. Te justification tat (3.1) interpolates is obvious: N p N (x j ) = N y k L k (x j ) = y k L k (x j ) = y j. k=0 k=0 It remains to see tat p N is te unique interpolating polynomial. For tis purpose, assume tat bot p N and q N take on te value y j at x j. Ten r N = p N q N is a polynomial of degree N tat as a root at eac of te N + 1 points x 0,..., x N. Te fundamental teorem of algebra, owever, says tat a nonzero polynomial of degree N can only ave N (complex) roots. Terefore, te only way for r N to ave N + 1 roots is tat it is te zero polynomial. So p N = q N. By definition, N p N (x) = f(x k )L k (x) k=0 is called te Lagrange interpolation polynomial of f at x j. Example 7. Linear interpolation troug (x 1, y 1 ) and (x 2, y 2 ): x x L 1 (x) = 2 x x 1, L 2 (x) =, x 1 x 2 x 2 x 1 p 1 (x) = y 1 L 1 (x) + y 2 L 2 (x) y 2 y 1 y 1 x 2 y 2 x 1 = x + x 2 x 1 x 2 x 1 y 2 = y 1 + y1 (x x 1 ). x 2 x 1 4

5 3.1. POLYNOMIAL INTERPOLATION Example 8. (Example (6.1) in Suli-Mayers) Consider f(x) = e x, and interpolate it by a parabola (N = 2) from tree samples at x 0 = 1, x 1 = 0, x 2 = 1. We build (x x 1 )( x x 2 ) 1 L 0 (x) = = x(x 1) (x 0 x 1 )(x 0 x 2 ) 2 Similarly, 1 L 1 (x) = 1 x 2, L 2 (x) = x(x + 1). 2 So te quadratic interpolant is p 2 (x) = e 1 L 0 (x) + e 0 L 1 (x) + e 1 L 2 (x), = 1 + sin(1) x + (cos(1) 1) x 2, x x 2. Anoter polynomial tat approximates e x reasonably well on [ 1, 1] is te Taylor expansion about x = 0: x 2 t 2 (x) = 1 + x +. 2 Manifestly, p 2 is not very different from t 2. (Insert picture ere) Let us now move on to te main result concerning te interpolation error of smoot functions. Teorem 5. Let f C N+1 [a, b] for some N > 0, and let {x j : j = 0,..., N} be a collection of disjoint reals in [a, b]. Consider p N te Lagrange interpolation polynomial of f at x j. Ten for every x [a, b] tere exists ξ(x) [a, b] suc tat f (N+1) (ξ(x)) f(x) p N (x) = π N+1 (x), (N + 1)! were N+1 π N+1 (x) = (x j=1 x ). An estimate on te interpolation error follows directly from tis teorem. Set ( M N+1 = max f N+1) (x) x [a,b] j 5

6 CHAPTER 3. INTERPOLATION (wic is well defined since f (N+1) is continuous by assumption, ence reaces its lower and upper bounds.) Ten MN+1 f(x) p N (x) (N + 1)! π N+1(x) In particular, we see tat te interpolation error is zero wen x = x j for some j, as it sould be. Let us now prove te teorem. Proof. (Can be found in Suli-Mayers, Capter 6) In conclusion, te interpolation error: depends on te smootness of f via te ig-order derivative f (N+1) ; as a factor 1/(N + 1)! tat decays fast as te order N ; and is directly proportional to te value of π N+1 (x), indicating tat te interpolant may be better in some places tan oters. Te natural follow-up question is tat of convergence: can we always expect convergence of te polynomial interpolant as N? In oter words, does te factor 1/(N + 1)! always win over te oter two factors? Unfortunately, te answer is no in general. Tere are examples of very smoot (analytic) functions for wic polynomial interpolation diverges, particularly so near te boundaries of te interplation interval. Tis beavior is called te Runge penomenon, and is usually illustrated by means of te following example. Example 9. (Runge penomenon) Let f(x) for x [ 5, 5]. Interpolate it at equispaced points x j = 10j/N, were j = N/2,..., N/2 and N is even. It is easy to ceck numerically tat te interpolant diverges near te edges of [ 5, 5], as N. See te Trefeten textbook on page 44 for an illustration of te Runge penomenon. (Figure ere) If we ad done te same numerical experiment for x [ 1, 1], te interpolant would ave converged. Tis sows tat te size of te interval matters. Intuitively, tere is divergence wen te size of te interval is larger tan te features, or caracteristic lengt scale, of te function (ere te widt of te bump near te origin.) 6

7 3.2. POLYNOMIAL RULES FOR INTEGRATION Te analytical reason for te divergence in te example above is due in no small part to te very large values taken on by π N+1 (x) far away from te origin in contrast to te relatively small values it takes on near te origin. Tis is a problem intrinsic to equispaced grids. We will be more quantitative about tis issue in te section on Cebysev interpolants, were a remedy involving non-equispaced grid points will be explained. As a conclusion, polynomial interpolants can be good for small N, and on small intervals, but may fail to converge (quite dramatically) wen te interpolation interval is large. 3.2 Polynomial rules for integration In tis section, we return to te problem of approximating b u(x)dx by a a weigted sum of samples u(x j ), also called a quadrature. Te plan is to form interpolants of te data, integrate tose interpolants, and deduce corresponding quadrature formulas. We can formulate rules of arbitrarily ig order tis way, altoug in practice we almost never go beyond order 4 wit polynomial rules Polynomial rules Witout loss of generality, consider te local interpolants of u(x) formed near te origin, wit x 0 = 0, x 1 = and x 1 =. Te rectangle rule does not belong in tis section: it is not formed from an interpolant. Te trapezoidal rule, were we approximate u(x) by a line joining (0, u(x 0 )) and (, u(x 1 )) in [0, ]. We need 2 derivatives to control te error: u (ξ(x)) u(x) = p 1 (x) + x(x ), 2 p 1 (x) = u(0)l 0 (x) + u()l 1 (x), 0 L 0 (x) = x, L x 1 (x) =, L 0 (x)dx = L 1 (x)dx = /2, (areas of triangles) 0 0 u (ξ(x)) x (x 2 ) dx C max u (ξ) 3. ξ 7

8 CHAPTER 3. INTERPOLATION Te result is 0 ( ) u(0) + u() u(x) dx = + O( 3 ). 2 As we ave seen, te terms combine as 1 N 1 u(x) dx = u(x 0 ) + u(x j 1) + u(x N ) + O( 2 ) j=1 Simpson s rule, were we approximate u(x) by a parabola troug (, u(x 1 )), (0, u(x 0 )), and (, u(x 1 )) in [, ]. We need tree derivatives to control te error: u (ξ(x)) u(x) = p 2 (x) + (x + )x(x ), 6 p 2 (x) = u( )L 1 (x) + u(0)l 0 (x) + u()l 1 (x), x(x ) (x + )(x ) (x + )x L 1 (x) =, L0 (x) =, L1 (x) =, L 1 (x)dx = L 1 (x)dx = /3, L 0 (x)dx = 4/3, u (ξ(x)) (x + )x(x ) dx C max u (ξ) Te result is ( ) u( ) + 4u(0) + u() u(x) dx = + O( 4 ). 3 Te composite Simpson s rule is obtained by using tis approximation on [0, 2] [2, 4]... [1 2, 1], adding te terms, and recognizing tat te samples at 2n (except 0 and 1) are represented twice. ( ) 1 u(0) + 4u() + 2u(2) + 4u(3) u(1 2) + 4u(1 ) + u(1) u(x)dx = 0 3 It turns out tat te error is in fact O( 5 ) on [, ], and O( 4 ) on [0, 1], a result tat can be derived by using symmetry considerations (or canceling te terms from Taylor expansions in a tedious way.) For tis, we need u to be four times differentiable (te constant in front of 5 involves u.) ξ 8

9 3.3. POLYNOMIAL RULES FOR DIFFERENTIATION Te iger-order variants of polynomial rules, called Newton-Cotes rules, are not very interesting because te Runge penomenon kicks in again. Also, te weigts (like 1,4,2,4,2, etc.) become negative, wic leads to unacceptable error magnification if te samples of u are not known exactly. Piecewise spline interpolation is not a good coice for numerical integration eiter, because of te two leftover degrees of freedom, wose arbitrary coice affects te accuracy of quadrature in an unacceptable manner. (We ll study splines in a later section.) We ll return to (useful!) integration rules of arbitrarily ig order in te scope of spectral metods. 3.3 Polynomial rules for differentiation A systematic way of deriving finite difference formulas of iger order is to view tem as derivatives of a polynomial interpolant passing troug a small number of points neigboring x j. For instance (again we use, 0, as reference points witout loss of generality): Te forward difference at 0 is obtained from te line joining (0, u(0)) and (, u()): p 1 (x) = u(0)l 0 (x) + u()l 1 (x), x x L 0 (x) =, L 1 (x) =, u() u(0) p 1(0) =. We already know tat u (0) p 1(0) = O(). Te centered difference at 0 is obtained from te line joining (, u( )) and (, u()) (a simple exercise), but it is also obtained from differentiating te parabola passing troug te points (, u( )), (0, u(0)), and (, u()). Indeed, p 2 (x) = u( )L 1 (x) + u(0)l 0 (x) + u()l 1(x), x(x ) (x + )(x ) (x + )x L 1 (x) =, L 0 (x) =, L 1 (x) =, x 2x p 2x + 2(x) = u( ) + u(0) + u(),

10 CHAPTER 3. INTERPOLATION u() ( p u ) 2 (0) =. 2 We already know tat u (0) p 2(0) = O( 2 ). Oter examples can be considered, suc as te centered second difference (-1 2-1), te one-sided first difference (-3 4-1), etc. Differentiating one-sided interpolation polynomials is a good way to obtain one-sided difference formulas, wic comes in andy at te boundaries of te computational domain. Te following result establises tat te order of te finite difference formula matces te order of te polynomial being differentiated. Teorem 6. Let f C N+1 [a, b], {x j, j = 0,..., N} some disjoint points, and p N te corresponding interpolation polynomial. Ten f (N+1) (ξ) f (x) p N(x) = π N (x), N! for some ξ [a, b], and were π N (x) = (x η 1 )... (x η N ) for some η j [x j 1, xj]. Te proof of tis result is an application of Rolle s teorem tat we leave out. (It is not simply a matter of differentiating te error formula for polynomial interpolation, because we ave no guarantee on dξ/dx.) A consequence of tis teorem is tat te error in computing te derivative is a O( N ) (wic comes from a bound on te product π N (x).) It is interesting to notice, at least empirically, tat te Runge s penomenon is absent wen te derivative is evaluated at te center of te interval over wic te interpolant is built. 3.4 Piecewise polynomial interpolation Te idea of piecewise polynomial interpolation, also called spline interpolation, is to subdivide te interval [a, b] into a large number of subintervals [x j 1, x j], and to use low-degree polynomials over eac subintervals. Tis elps avoiding te Runge penomenon. Te price to pay is tat te interpolant is no longer a C function instead, we lose differentiability at te junctions between subintervals, were te polynomials are made to matc. 10

11 3.4. PIECEWISE POLYNOMIAL INTERPOLATION If we use polynomials of order n, ten te interpolant is piecewise C, and overall C k, wit k n 1. We could not expect to ave k = n, because it would imply tat te polynomials are identical over eac subinterval. We ll see two examples: linear splines wen n = 1, and cubic splines wen n = 3. (We ave seen in te omework wy it is a bad idea to coose n = 2.) Linear splines We wis to interpolate a continuous function f(x) of x [a, b], from te knowledge of f(x j ) at some points x j, j = 0,..., N, not necessarily equispaced. Assume tat x 0 = a and x N = b. Te piecewise linear interpolant is build by tracing a straigt line between te points (x j 1, f(x j 1 )) and (x j, f(x j ))); for j = 1,..., N te formula is simply x j 1 x x x j 1 s L (x) = f(xj 1 ) + f(x j ), x [x j 1, x j]. x j 1 x j x 1 x j In tis case we see tat s L (x) is a continuous function of x, but tat its derivative is not continuous at te junction points, or nodes x j. If te function is at least twice differentiable, ten piecewise linear interpolation as second-order accuracy, i.e., an error O( 2 ). Teorem 7. Let f C 2 [a, b], and let = max j=1,...,n (x j x j 1 ) be te grid diameter. Ten 2 f s L L [a,b] 8 f L [a,b]. Proof. Let x [x j 1, x j ] for some j = 1,..., N. We can apply te basic result of accuracy of polynomial interpolation wit n = 1: tere exists ξ [x j 1, x j ] suc tat 1 f(x) s L (x) = f (ξ) (x x j 1)(x x j ), x [x j 1, x j ]. 2 Let j = x j x j 1. It is easy to ceck tat te product (x x j 1 )(x x j ) xj 1+x takes its maximum value at te midpoint j, and tat te value is 2 2 j/4. We ten ave 2 j f(x) s L (x) j max f (ξ), x [x j 1, x j ]. 8 ξ [x j 1,x j ] Te conclusion follows by taking a maximum over j. 11

12 CHAPTER 3. INTERPOLATION We can express any piecewise linear interpolant as a superposition of tent basis functions φ k (x): s L (x) = c k φ k (x), k were φ k (x) is te piecewise linear function equal to 1 at x = x k, and equal to zero at all te oter grid points x j, j = k. Or in sort, φ k (x j ) = δ jk. On an equispaced grid x j = j, an explicit formula is were 1 φ k (x) = S (1) (x x k ), S (1) (x) = x + 2(x ) + + (x 2) +, and were x + denotes te positive part of x (i.e., x if x 0, and zero if x < 0.) Observe tat we can simply take c k = f(x k ) above. Te situation will be more complicated wen we pass to iger-order polynomials Cubic splines Let us now consider te case n = 3 of an interpolant wic is a tird-order polynomial, i.e., a cubic, on eac subinterval. Te most we can ask is tat te value of te interpolant, its derivative, and its second derivative be continuous at te junction points x j. Definition 5. (Interpolating cubic spline) Let f C[a, b], and {x j, j = 0,..., N} [a, b]. An interpolating cubic spline is a function s(x) suc tat 1. s(x j ) = f(x j ); 2. s(x) is a polynomial of degree 3 over eac segment [x j 1, x j ]; 3. s(x) is globally C 2, i.e., at eac junction point x j, we ave te relations s(x j ) = s(x+ j ), s (x j ) = s (x + j ), s (x j ) = s (x + j ), were te notations s(x j ) and s(x+ j ) refer to te adequate limits on te left and on te rigt. 12

13 3.4. PIECEWISE POLYNOMIAL INTERPOLATION Let us count te degrees of freedom. A cubic polynomial as 4 coefficients. Tere are N + 1 points, ence N subintervals, for a total of 4N numbers to be specified. Te interpolating conditions s(x j ) = f(x j ) specify two degrees of freedom per polynomial: one value at te left endpoint x j 1, and one value at te rigt endpoint x j. Tat s 2N conditions. Continuity of s(x) follows automatically. Te continuity conditions on s, s are imposed only at te interior grid points x j for j = 1,..., N 1, so tis gives rise to 2(N 1) additional conditions, for a total of 4N 2 equations. Tere is a mismatc between te number of unknowns (4N) and te number of conditions (4N 2). Two more degrees of freedom are required to completely specify a cubic spline interpolant, ence te precaution to write an interpolant in te definition above, and not te interpolant. Te most widespread coices for fixing tese two degrees of freedom are: Natural splines: s (x 0 ) = s (x N ) = 0. If s(x) measures te displacement of a beam, a condition of vanising second derivative corresponds to a free end. Clamped spline: s (x 0 ) = p 0, s (x N ) = p N, were p 0 and p N are specified values tat depend on te particular application. If s(x) measures te displacement of a beam, a condition of vanising derivative corresponds to a orizontal clamped end. Periodic spline: assuming tat s(x 0 ) = s(x N ), ten we also impose s (x 0 ) = s (x N ), s (x 0 ) = s (x N ). Let us now explain te algoritm most often used for determining a cubic spline interpolant, i.e., te 4N coefficients of te N cubic polynomials, from te knowledge of f(x j ). Let us consider te natural spline. It is advantageous to write a system of equations for te second derivatives at te grid points, tat we denote σ j = s (x j ). Because s(x) is piecewise cubic, we know tat s (x) is piecewise linear. Let j = x j x j 1. We can write x s j x x x j 1 (x) = σ j 1 + σ j, x [x j 1, x j ]. j j 13

14 CHAPTER 3. INTERPOLATION Hence (x j x) 3 (x s(x) = σ j 1 + x j 1) 3 σ j + α j (x x j 1 ) + βj(x j x). 6 j 6 j We could ave written ax + b for te effect of te integration constants in te equation above, but writing it in terms of α j and β j makes te algebra tat follows simpler. Te two interpolation conditions for [x j 1, x j ] can be written σj 1 2 j s(x j 1 ) = f(x j 1 ) f(x 1 ) = j + β 6 j j, σj 2 j s(x j ) = f(x j ) f(x j ) = + j α j. 6 One ten isolates α j, β j in tis equation; substitutes tose values in te equation for s(x); and evaluates te relation s (x j ) = s (x + j ). Given tat σ 0 = σ N = 0, we end up wit a system of N 1 equations in te N 1 unknowns σ 1,..., σ N. Skipping te algebra, te end result is ( ) f(x j+1 ) f(x j ) f(x j ) f(x j 1 ) j σ j 1 + 2( j+1 + j )σ j + j+1 σ j+1 = 6. j+1 We are in presence of a tridiagonal system for σ j. It can be solved efficiently wit Gaussian elimination, yielding a LU decomposition wit bidiagonal factors. Unlike in te case of linear splines, tere is no way around te fact tat a linear system needs to be solved. Notice tat te tridiagonal matrix of te system above is diagonally dominant (eac diagonal element is strictly greater tan te sum of te oter elements on te same row, in absolute value), ence it is always invertible. One can ceck tat te interpolation for cubic splines is O( 4 ) well away from te endpoints. Tis requires an analysis tat is too involved for te present set of notes. Finally, like in te linear case, let us consider te question of expanding a cubic spline interpolant as a superposition of basis functions s(x) = c k φ k (x). k Tere are many ways of coosing te functions φ k (x), so let us specify tat tey sould ave as small a support as possible, tat tey sould ave te j 14

15 3.4. PIECEWISE POLYNOMIAL INTERPOLATION same smootness C 2 as s(x) itself, and tat tey sould be translates of eac oter wen te grid x j is equispaced wit spacing. Te only solution to tis problem is te cubic B-spline. Around any interior grid point x k, is supported in te interval [x k 2, x k+2 ], and is given by te formula 1 φ k (x) = S (3) (x x 4 3 k 2 ), were S (x) = x 3 3 (3) + 4(x ) + + 6(x 2) 3 + 4(x 3) (x 4) 3 +, and were x + is te positive part of x. One can ceck tat φ k (x) takes te value 1 at x k, 1/4 at x k ± 1, zero outside of [xk 2, x k+2], and is C 2 at eac junction. It is a bell-saped curve. it is an interesting exercise to ceck tat it can be obtained as te convolution of two tent basis functions of linear interpolation: S (3) (x) = cs (1) (x) S (1) (x), were c is some constant, and te symbol denotes convolution: f g(x) = f(y)g(x y) dy. R Now wit cubic B-splines, we cannot put c k = f(x k ) anymore, since φ k (x j ) = δ jk. Te particular values of c k are te result of solving a linear system, as mentioned above. Again, tere is no way around solving a linear system. In particular, if f(x k ) canges at one point, or if we add a datum point (x k, f k ), ten te update requires re-computing te wole interpolant. Canging one point as ripple effects trougout te wole interval [a, b]. Tis beavior is ultimately due to te more significant overlap between neigboring φ k in te cubic case tan in te linear case. 15

16 MIT OpenCourseWare ttp://ocw.mit.edu Introduction to Numerical Analysis Spring 2012 For information about citing tese materials or our Terms of Use, visit: ttp://ocw.mit.edu/terms.

Finite Difference Approximations

Finite Difference Approximations Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

Verifying Numerical Convergence Rates

Verifying Numerical Convergence Rates 1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

More information

Trapezoid Rule. y 2. y L

Trapezoid Rule. y 2. y L Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

More information

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

More information

SAT Subject Math Level 1 Facts & Formulas

SAT Subject Math Level 1 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

More information

Differentiable Functions

Differentiable Functions Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.

More information

Understanding the Derivative Backward and Forward by Dave Slomer

Understanding the Derivative Backward and Forward by Dave Slomer Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

More information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

More information

Finite Volume Discretization of the Heat Equation

Finite Volume Discretization of the Heat Equation Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x

More information

Chapter 7 Numerical Differentiation and Integration

Chapter 7 Numerical Differentiation and Integration 45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

More information

ACT Math Facts & Formulas

ACT Math Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

More information

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

More information

Optimal Pricing Strategy for Second Degree Price Discrimination

Optimal Pricing Strategy for Second Degree Price Discrimination Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases

More information

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry. Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

More information

Instantaneous Rate of Change:

Instantaneous Rate of Change: Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

More information

since by using a computer we are limited to the use of elementary arithmetic operations

since by using a computer we are limited to the use of elementary arithmetic operations > 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

More information

Exercises for numerical integration. Øyvind Ryan

Exercises for numerical integration. Øyvind Ryan Exercises for numerical integration Øyvind Ryan February 25, 21 1. Vi ar r(t) = (t cos t, t sin t, t) Solution: Koden blir % Oppgave.1.11 b) quad(@(x)sqrt(2+t.^2),,2*pi) a. Finn astigeten, farten og akselerasjonen.

More information

Math 113 HW #5 Solutions

Math 113 HW #5 Solutions Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

More information

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.) Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

More information

7.6 Complex Fractions

7.6 Complex Fractions Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

More information

1 Derivatives of Piecewise Defined Functions

1 Derivatives of Piecewise Defined Functions MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

More information

ACTIVITY: Deriving the Area Formula of a Trapezoid

ACTIVITY: Deriving the Area Formula of a Trapezoid 4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

More information

a = x 1 < x 2 < < x n = b, and a low degree polynomial is used to approximate f(x) on each subinterval. Example: piecewise linear approximation S(x)

a = x 1 < x 2 < < x n = b, and a low degree polynomial is used to approximate f(x) on each subinterval. Example: piecewise linear approximation S(x) Spline Background SPLINE INTERPOLATION Problem: high degree interpolating polynomials often have extra oscillations. Example: Runge function f(x) = 1 1+4x 2, x [ 1, 1]. 1 1/(1+4x 2 ) and P 8 (x) and P

More information

New Vocabulary volume

New Vocabulary volume -. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

More information

Geometric Stratification of Accounting Data

Geometric Stratification of Accounting Data Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

More information

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution 1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

More information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)

More information

Surface Areas of Prisms and Cylinders

Surface Areas of Prisms and Cylinders 12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

More information

Proof of the Power Rule for Positive Integer Powers

Proof of the Power Rule for Positive Integer Powers Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

More information

The EOQ Inventory Formula

The EOQ Inventory Formula Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

More information

Piecewise Cubic Splines

Piecewise Cubic Splines 280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computer-assisted design), CAM (computer-assisted manufacturing), and

More information

2 Limits and Derivatives

2 Limits and Derivatives 2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

More information

CHAPTER 7. Di erentiation

CHAPTER 7. Di erentiation CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

More information

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1) Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

More information

Exam 2 Review. . You need to be able to interpret what you get to answer various questions.

Exam 2 Review. . You need to be able to interpret what you get to answer various questions. Exam Review Exam covers 1.6,.1-.3, 1.5, 4.1-4., and 5.1-5.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam

More information

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

3.5 Spline interpolation

3.5 Spline interpolation 3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly non-locally.

More information

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

More information

Tangent Lines and Rates of Change

Tangent Lines and Rates of Change Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

More information

Natural cubic splines

Natural cubic splines Natural cubic splines Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology October 21 2008 Motivation We are given a large dataset, i.e. a function sampled

More information

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctions-intro.tex October, 9 Note tat tis section of notes is limitied to te consideration

More information

i = 0, 1,, N p(x i ) = f i,

i = 0, 1,, N p(x i ) = f i, Chapter 4 Curve Fitting We consider two commonly used methods for curve fitting, namely interpolation and least squares For interpolation, we use first polynomials then splines Then, we introduce least

More information

Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction

Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.

More information

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set

More information

Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

More information

Solution Derivations for Capa #7

Solution Derivations for Capa #7 Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park

More information

6. Differentiating the exponential and logarithm functions

6. Differentiating the exponential and logarithm functions 1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

More information

Research on the Anti-perspective Correction Algorithm of QR Barcode

Research on the Anti-perspective Correction Algorithm of QR Barcode Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

More information

Math Warm-Up for Exam 1 Name: Solutions

Math Warm-Up for Exam 1 Name: Solutions Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warm-up to elp you begin studying. It is your responsibility to review te omework problems, webwork

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Math Test Sections. The College Board: Expanding College Opportunity

Math Test Sections. The College Board: Expanding College Opportunity Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

More information

5 Numerical Differentiation

5 Numerical Differentiation D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives

More information

Theoretical calculation of the heat capacity

Theoretical calculation of the heat capacity eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

More information

Discovering Area Formulas of Quadrilaterals by Using Composite Figures

Discovering Area Formulas of Quadrilaterals by Using Composite Figures Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will

More information

Projective Geometry. Projective Geometry

Projective Geometry. Projective Geometry Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

More information

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade? Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

More information

Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

is called the geometric transformation. In computers the world is represented by numbers; thus geometrical properties and transformations must

is called the geometric transformation. In computers the world is represented by numbers; thus geometrical properties and transformations must Capter 5 TRANSFORMATIONS, CLIPPING AND PROJECTION 5.1 Geometric transformations Tree-dimensional grapics aims at producing an image of 3D objects. Tis means tat te geometrical representation of te image

More information

A Low-Temperature Creep Experiment Using Common Solder

A Low-Temperature Creep Experiment Using Common Solder A Low-Temperature Creep Experiment Using Common Solder L. Roy Bunnell, Materials Science Teacer Soutridge Hig Scool Kennewick, WA 99338 roy.bunnell@ksd.org Copyrigt: Edmonds Community College 2009 Abstract:

More information

Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012 Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

More information

Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) + Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

More information

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q = Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))

More information

The modelling of business rules for dashboard reporting using mutual information

The modelling of business rules for dashboard reporting using mutual information 8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

More information

Area of a Parallelogram

Area of a Parallelogram Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make

More information

ME422 Mechanical Control Systems Modeling Fluid Systems

ME422 Mechanical Control Systems Modeling Fluid Systems Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

More information

SAT Math Facts & Formulas

SAT Math Facts & Formulas Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

More information

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,

More information

CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

More information

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53 ontents 4 Trigonometry 4.1 Rigt-angled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65

More information

Reinforced Concrete Beam

Reinforced Concrete Beam Mecanics of Materials Reinforced Concrete Beam Concrete Beam Concrete Beam We will examine a concrete eam in ending P P A concrete eam is wat we call a composite eam It is made of two materials: concrete

More information

OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS

OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

More information

Writing Mathematics Papers

Writing Mathematics Papers Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

More information

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly non-contiguous or consisting

More information

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

More information

1 Review of Least Squares Solutions to Overdetermined Systems

1 Review of Least Squares Solutions to Overdetermined Systems cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

More information

Area of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2

Area of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2 Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

More information

The differential amplifier

The differential amplifier DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c

More information

(Refer Slide Time: 00:00:56 min)

(Refer Slide Time: 00:00:56 min) Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

More information

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3 LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We

More information

2.28 EDGE Program. Introduction

2.28 EDGE Program. Introduction Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

More information

2.1: The Derivative and the Tangent Line Problem

2.1: The Derivative and the Tangent Line Problem .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

More information

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly non-contiguous or consisting

More information

Numerical Analysis An Introduction

Numerical Analysis An Introduction Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs

More information

November 16, 2015. Interpolation, Extrapolation & Polynomial Approximation

November 16, 2015. Interpolation, Extrapolation & Polynomial Approximation Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic

More information

Sum of the generalized harmonic series with even natural exponent

Sum of the generalized harmonic series with even natural exponent Rendiconti di Matematica, Serie VII Volume 33, Roma (23), 9 26 Sum of te generalized armonic erie wit even natural exponent STEFANO PATRÌ Abtract: In ti paper we deal wit real armonic erie, witout conidering

More information