Numerical Analysis Formulae Booklet
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1 Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres Fte Dfferece Approxmatos Egevalues ad Egevectors of Matrces Vector ad Matrx Norms Ital Value Problem solvers Polyomal Iterpolato Secod Order Dfferece Equatos Euler s Formula ad oter Trgoometrc Idettes Tomas Algortm (TDMA) Newto Rapso Sceme Lax-Wedroff Sceme Crak-Ncolso sceme b UFMFX Formula Booklet Page of 9
2 . Iteratve Scemes for Systems of Lear Algebrac Equatos:. Successve Over-Relaxato (SOR) terato sceme for solvg were s a matrx: ( ). Te Jacob terato matrx for solvg s gve by (F.) were, wt beg a dagoal matrx, ad are strctly lower ad strctly upper tragular matrces, respectvely.. Taylor Seres (a) Te Taylor seres expaso of a fucto f ( x ) about te pot 0 as te form f ( x ) f ( x) f ( x) f ( x) f ( x)! 3!! were x p x. 3 f ( ) ( x) f ( )! ( ) ( p) (F.) (b) Te Taylor seres expaso of a fucto f ( x, y k) about te pot defed by 0, k 0 as te form up to ad cludg secod order terms of f ( x, y k) f f x kf y! f xx kf xy k! f yy (F.) UFMFX Formula Booklet Page of 9
3 3. Fte Dfferece Approxmatos (a) Regular grd: u x u u cetral dfferece (F3.) u x u u forward dfferece (F3.) u u u u x cetral dfferece (F3.3) (b) Irregular grd: 0 x x u x x x u ( ) u u ( ) x (F3.4) u x u ( ) u u ( ) (F3.5) 4. Egevalues ad Egevectors of Matrces. (a) Egevalues of commo trdagoal system Te egevalues of a commo m m trdagoal matrx: a b c a b c a b c a b c a as m dstct egevalues gve by: s s a bc cos s,, m m (F4.) UFMFX Formula Booklet Page 3 of 9
4 (b) () Commo egevectors If matrx A ad B sare commo egevectors te tese egevectors are also egevectors k A k B for ay scalars k, k. of () If matrx A s vertble, te A ad A sare te same egevectors. (c) Gersgor s Crcle Teorem Gve a matrx A, te all ts egevalues le te rego D D were D z : z a R ad R j j j a. (F4.) (d) Oter propertes regardg egevalues of matrces () All te egevalues of a real symmetrc matrx are real. () If matrx A s a real trdagoal matrx ad all ts o-dagoal etres are of te same sg (eter all egatve or all postve), te all ts egevalues are real. UFMFX Formula Booklet Page 4 of 9
5 5. Vector ad Matrx Norms A vector orm of a vector deoted by, s a real-valued fucto of tat satsfes te followg propertes: s ay scalar (te dot deotg multplcato) te tragle equalty Some commo vector orms are gve below for a vector : u u u u m (F5.) u u u u m (F5.) max,,, m u u u u (F5.3) Gve a matrx ad gve a vector orm, te te duced orm of te matrx, deoted by, s defed as: Usg te defto of te duced matrx orm (F5.4), te followg matrx orms (of matrces wt real valued etres) result from te vector orms defed (F5.)-(F5.3), represetg te spectral radus (F5.5) UFMFX Formula Booklet Page 5 of 9
6 6. Ital Value Problem solvers For te frst order tal value problem y f ( t,, y(0) y 0 6. Euler s Metod y y f 6. Te Modfed Euler Metod y y f P y y f f P 6.3 Te Classcal d Order Ruge-Kutta Metod k f t, y k f t, y k y y k 6.4 Te Classcal 4 t Order Ruge-Kutta Metod k f t, y 4 3 k f t, y k k3 f t, y k k f t, y k y y k k k k I te above expressos f f t, y ). ( UFMFX Formula Booklet Page 6 of 9
7 7. Polyomal Iterpolato Te polyomal of degree k passg troug k pots ( t, f), ( t, f ), ( t, f ),... s gve by: p ( t) f rf r( r ) f r( r )( r ) f... r( r )...( r k ) k f (F7.) 3 k! 3! ( k)! were t t r, f f f, k k f t, y( t) pk ( t) r( r )( r )...( r k ) f ( c), t c t. (F7.) k! ad 8. Secod Order Dfferece Equatos Te costat coeffcet secod order dfferece equato a y b y c y 0 were a, b ad c are costats, as a geeral soluto, provded p p, gve by were p ad p satsfy te auxlary equato y Ap Bp ap bp c 0 Teorem: Te roots of te quadratc equato are bouded by oe modulus f ad oly f all te followg codtos old: Q() 0, Q( ) 0, c a. 9. Euler s Formula ad oter Trgoometrc Idettes Euler s Formula: e cos s (F9.) cos s cos (F9.) UFMFX Formula Booklet Page 7 of 9
8 0. Tomas Algortm (TDMA) Cosder te followg trdagoal system of lear equatos, were te v s are te ukows: b c v d a b c 0 0 v d 0 a3 b3. 0 v 3 d cm am b m v m d m (F0.) Te TDMA states tat te soluto to ts trdagoal system s also a system to te followg bdagoal system of lear equatos: ˆ cˆ v d 0 cˆ 0 0 v ˆ d v ˆ 3 d3 (F0.) 0 0. cˆ m v m dˆ m were te forward step s gve by te followg: c cˆ, b c cˆ for,.., m b cˆ a ˆ, b d d d dˆ a d m ˆ for,.., b ˆ c a Te backward step, to obta te values of v, s to perform back substtuto o (F0.). UFMFX Formula Booklet Page 8 of 9
9 . Newto Rapso Sceme Cosder te followg system of m omogeeous olear equatos: f U, U,..., U 0,,..., m. (F.) m Te te Newto Rapso metod for solvg te system (F.) s a teratve sceme as follows: U U ( p) ( p) ( p) were te colum matrx A ( p ) f were A s te mm matrx gve by: A f ( p) s a soluto to te followg lear system of equatos: j, wt te etres evaluated at te p t terato stage U j ad te colum matrx f also as ts etres evaluated at te p t terato stage.. Lax-Wedroff Sceme Te Lax-Wedroff metod for te umercal soluto of equatos of te form: s gve by: were (F.) ( ) (F.) were, for example,. Wt ( ) ( ) (F.3) ( ) ( ) (F.4) 3. Crak Ncolso Sceme Te Crak-Ncolso sceme for te approxmato to te partal dfferetal equato gve by: s ( ) (( ) ( )) UFMFX Formula Booklet Page 9 of 9
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