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# Numerical Analysis MATH 640

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This 8 page Class Notes was uploaded by Alphonso Thompson on Thursday October 29, 2015. The Class Notes belongs to MATH 640 at Western Carolina University taught by Erin McNelis in Fall. Since its upload, it has received 12 views. For similar materials see /class/230960/math-640-western-carolina-university in Mathematics (M) at Western Carolina University.

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Date Created: 10/29/15

MATH 640 7 Numerical Analysis Section 66 Special Types of Matrices De nition 1 Strictly Diagonally Dominant The n x n matrix A is said to be strictly diagonally dominant when 71 laiilgt 2 Mil j1i holds for each i 1 n Theorem 1 A strictly diagonally dominant matrix A is nonsingular Moreover in this case Gaussian elimination can be performed on any linear system of the form AX b to obtain its unique solution without row or column interchanges and the computations will be stable with respect to the growth of roundo errors De nition 2 Symmetric Positive De nite A matrix A is positive de nite it is symmetric and if XTAX gt 0 for euery n dimensional uector X 0 Theorem 2 IfA is an n X n positiue de nite matrix then 1 A is nonsingular 2 am gt0for eachi12n lt 5 W i W 4 aij2 lt aiiajj for each i De nition 3 Leading Principal Submatrix A leading principal submatrix of a matrix A is a matrix of the form all 03912 i i alk 121 122 39 39 azk Ak am akz akk forsomelgkgn Theorem 3 A symmetric matrix A is positiue de nite if and only if each of its leading principal submatrices has a positiue determinant Theorem 4 The symmetric matrix A is positiue de nite if an only if Gaussian elimination without row interchanges can be performed on the linear system AX b with all piuot elements positiue Moreover in this case the computations are stable with respect to the growth of roundo errors Corollary 1 The matrix A is positive de nite if and only if A can be factored in the form LDLT7 where L is lower triangular with 1 s on the diagonal and D is a diagonal matrix with positive diagonal entries Corollary 2 The matrix A is positive de nite if and only if A can be factored in the form LLT7 where L is lower triangular with nonzero diagonal entries LDLT Factorization To factor the positive de nite n x n matrix A into the form LDLT7 where L is a lower triangular matrix with 1 s along the diagonal and D is a diagonal matrix with positive entries on the diagonal Input the dimension7 n matrix A ah7 where 1 g i g n and 1 g j g n Output the entries Z777 for 1 g j lt i and 1 g i g n ofL and 17 for 1 g i g n of D Step1 Fori177ndoSteps2 4 Step 2 For j 177i7 17 set vj lijdj Step 32 Set an 7 lij Uj 13971 Step 4 For i 17 7n 7 set ljl39 aji 7 lekvk di k1 Step 5 OUTPUTUH for j 177i7 1 and i 177n OUTPUTd7 for i 17 7n STOP Corollary 3 Let A be a symmetric n X n matrix for which Gaussian elimination can be applied without row interchanges Then A can be factored into LDLT7 where L is lower triangular with 1 s on its diagonal and D is the diagonal matrix with a111 7 may on its diagonal Choleski7s Factorization To factor the positive de nite n x n matrix A into the form LLT7 where L is a lower triangular ma trix Input the dimension7 n matrix A aij7 where 1 g i g n and 1 g j g n Output the entries Z777 for 1 g j lt i and 1 g i g n of L Step 12 Set l11 all Step 2 For j 27 7n7 set ljl ajlln Step3 Fori277n717doSteps4and5 1 Step 4 Set 177 1777 21 k1 iil Step 52 For l 17 77l 7 set ljl39 07139 7 k1 n71 Step 6 Set 1 am 7 21377 k1 Step 7 OUTPUTUH for j 177i7 1 and i 177n STOP De nition 4 Band Matrix An n x n matrix is called a band matrix if integers p and q with 1 lt 197 q lt it exist with the property that aij 0 wheneverp j 7i or q i 7 j The bandwidth of a band matrix is de ned as w p 171 De nition 5 Tridiagonal Matrix Matrices of bandwidth 5 are called tridiagonal Tridiagonal matrices can be decomposed as A LU where L is lower diagonal with and U is upper diagonal with the forms L 11 0 0 0 1 an 0 0 21 22 0 0 0 1 U23 0 0 32 33 0 0 0 0 1 U34 0 0 0 and R 0 0 j 39 39 39 O 39 39 unilm 0 0 ln il nn 0 0 0 Crout Factorization for Tridiagonal Systems Purpose To solve the n x n linear system E1 3 a11951 a12952 a1n1 E2 3 121951 122952 123953 a2n1 Emil an71ri72ri72 artilmilmriil aniiy n anil L Fl Eri 3 arim71ri71 l armri anri1 Input the dimension7 n matrix A aij7 where 1 g i g n and 1 g j g n Output the solution x1 7xn Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Set 11 all U12 112111 2 1 a1n1lll39 Fori 2 777 7178813113141 amil lii aii lii71Ui71i Uii1 aii1lii Zi airi1 lii712i71lii set lnmil amnili rm arm 7 l Lilu il Li Zn anri1 7 l Lilz ill i Set xn 2n Forin711 7 set xi 2i7uii1xi1 OUTPUTx1x27 7xn STOP MATH 640 7 Numerical Analysis Section 66 Special Types of Matrices Wednesday January 28 2004 De nition 1 Strictly Diagonally Dominant The n x n matrix A is said to be strictly diagonally dominant when 71 laiilgt 2 Mil j1i holds for each i 1 n Theorem 1 A strictly diagonally dominant matrix A is nonsingular Moreover in this case Gaussian elimination can be performed on any linear system of the form AX b to obtain its unique solution without row or column interchanges and the computations will be stable with respect to the growth of roundo errors De nition 2 Symmetric Positive De nite A matrix A is positive de nite it is symmetric and if XTAX gt 0 for euery n dimensional uector X 0 Theorem 2 IfA is an n X n positiue de nite matrix then 1 A is nonsingular 239 aii gt0for eachi12n lt 5 W e W 4 aij2 lt aiiajj for each i De nition 3 Leading Principal Submatrix A leading principal submatrix of a matrix A is a matrix of the form all 03912 39 39 39 alk 121 122 39 39 azk Ak am akz akk forsomelgkgn Theorem 3 A symmetric matrix A is positiue de nite if and only if each of its leading principal submatrices has a positiue determinant Theorem 4 The symmetric matrix A is positiue de nite if an only if Gaussian elimination without row interchanges can be performed on the linear system AX b with all piuot elements positiue Moreover in this case the computations are stable with respect to the growth of roundo errors Corollary 1 The matrix A is positive de nite if and only if A can be factored in the form LDLT7 where L is lower triangular with 1 s on the diagonal and D is a diagonal matrix with positive diagonal entries Corollary 2 The matrix A is positive de nite if and only if A can be factored in the form LLT7 where L is lower triangular with nonzero diagonal entries LDLT Factorization To factor the positive de nite n x n matrix A into the form LDLT7 where L is a lower triangular matrix with 1 s along the diagonal and D is a diagonal matrix with positive entries on the diagonal Input the dimension7 n matrix A ah7 where 1 g i g n and 1 g j g n Output the entries Z777 for 1 g j lt i and 1 g i g n ofL and 17 for 1 g i g n of D Step1 Fori177ndoSteps2 4 Step 2 For j 17 7n7 set vj lijdj Step 32 Set an 7 lij Uj 13971 Step 4 For i 17 7n 7 set ljl39 aji 7 lekvk di k1 Step 5 OUTPUTUH for j 177i7 1 and i 177n OUTPUTd7 for i 17 7n STOP Corollary 3 Let A be a symmetric n X n matrix for which Gaussian elimination can be applied without row interchanges Then A can be factored into LDLT7 where L is lower triangular with 1 s on its diagonal and D is the diagonal matrix with a111 7 may on its diagonal Choleski7s Factorization To factor the positive de nite n x n matrix A into the form LLT7 where L is a lower triangular ma trix Input the dimension7 n matrix A aij7 where 1 g i g n and 1 g j g n Output the entries Z777 for 1 g j lt i and 1 g i g n of L Step 12 Set l11 all Step 2 For j 27 7n7 set ljl ajlln Step3 Fori277n717doSteps4and5 139 Step 4 Set lg k1 i71 Step 52 For l 17 77l 7 set ljl39 07139 7 k1 n71 Step 6 Set 1 am 7 21377 k1 Step 7 OUTPUTUH for j 177i7 1 and i 177n STOP MATH 640 7 Numerical Analysis Section 85 Trigonometric Polynomial Approximation 01 Fourier Polynomials and Series For Continuous Approach Recall the key idea from Section 83 Continuous Least Squares Approximation If we have an orthogonal set of functions polynomial gt0 gt1 gtn on a b with respect to the weight function wz the least squares polynomial approximation to f on a b is n k0 where for each h 01 n a mwmmomibwmm mm k wmwmmwz aka H mw where b wmwmwm 1 THIS HELD FOR ANY ORTHOGONAL FUNCTIONS NOT JUST POLYNOMIALSHH De nition 1 Set of Trigonometric Polynomials of Degree n Let L denote the set of all linear combinations of the functions gt0 gt1 gt2n1 where gt0 gtk COSkz for k 12771 gtnk sinkai for k 12771 1 le NOTE39 gt0 gt1 gt2n1 are an orthogonal set offunctions on 77r7r with respect to the weight function wm 1 The set is called the set of trigonometric polynomials of degree less than or equal to n and is denoted by L Then we can use this set of polynomials to get our approximating function call it Snz where M Sim WWW bk gtnk w H o D mmnn 2 Cammns Last Sqnaxes mgmmemc Appzaxxmzbmn The Foumr polynomaz not 272 book s 297m m m denoted 542 1 gram w sea 7quot 0 6Nm Wemslc bksn1 z e We a WNW w bk WWW Weem bu 0 w w He Oliver 21722542 meme uuMmlyto mpmw 19 542 1 We 272 54 ze thedtze Foumr SawsM n 2 Discrete Least Squares I xlgunumehxc Appxuxxmahun w em mumxsmae Agam mvmby Ema web when Zmisewf MWWPMem r71 0 n gt o o o o o o o lt x xmel xm xnul XIIIJ x2quot nmfaxmlv dmxbnbed muslin naval embed we hewe impmmse tm hy byx meme Wh 9wemnwnm 443w fur L We Eefme we an deemme nee emeweee one web we um mxmmxze ll39nsermze we need e aeme em 31mm Meme 5 Onhaganal Rmcmans e wee eqwttyxpamd wmqmgquot WM 1 1 immanww wow Lemma 1 If the integer r is not a multiple of 2m then 2m71 2m71 Z cosrzj 0 and Z sinrmj 0 j0 739 0 Moreouer ifr is not a multiple of m then 2m71 27ml 2 cosrzj2 m and Z 513T7 2 m j0 7390 Theorem 2 Discrete Least Squares Trigonometric Approximation The constants in the summation n71 Sm ancosnz 2akcoskz bksinkm k1 that minimize the least squares sum 2m71 Ea07a17 397an7b17 7bn71 Z W Sn90j2 j0 are 1 2m 1 ak i fzcoskmdz fork01n m F0 1 2m 1 bk i fzsinkxdm fork12n71 m

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