x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

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NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].

1 NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat are te oter ames for iteratio metod? (i Idirect metod (ii Successive approimatio metod. 3. State te coditio for te covergece of iteratio metod? '( i [a,b]. 4. Wat is te order of covergece of fied poit iteratio metod? Te order of covergece of fied poit iteratio metod is. Tat is, te covergece is liear. 5. Give te formula for iteratio metod. ( 6. State te coditio for te covergece of Newto -Rapso metod? f ( f "( f '( 7. Wat is te order of covergece for Newto-Rapso metod? Te order of covergece for Newto-Rapso metod is. 8. Write te formula for Newto-Rapso metod. f( f '( f ( N f '( By Newto-Rapso formula f ( ( N f '( N N 0. Fid a iterative formula to fid pt root of N usig Newto-Rapso metod. Let = p N. Te p p N N 0 ( p p f N f '( p By Newto-Rapso formula p f ( ( N f '( p p p p p N p p p ( p N p p. Derive a iterative formula to fid cube root of k usig Newto-Rapso metod. Let = 3 k ( k Te k k 0 3 f ( k f '( 3 By Newto-Rapso formula 9. Fid a iterative formula to fid N usig Newto-Rapso metod. Let = N. Te N N 0

2 3 f N f '( p p 3 N 3 ( ( N. Derive a iterative formula to fid te reciprocal of a positive umber usig Newto-Rapso metod. Let = N. Te N 0 N f ( N f '( By Newto-Rapso formula ( N f ( f '( N N = N ( N ( N 3. Write te direct metods to solve te system of equatios. (i Gauss Elimiatio metod (ii Gauss Jorda metod 4. Write te idirect metods (or iterative metods to solve te system of equatios. (i Gauss Jacobi metod ad (ii Gauss-Seidel metod 5. Distiguis betwee direct metods ad idirect metods of solvig a system of equatios AX=B. Direct metods Idirect metods Tey give oly te approimate Tey give eact value. value. Simple ad take less time. Tey are time cosumig. 6. Eplai briefly Gauss Elimiatio metod. I Gauss Elimiatio metod, te coefficiet matri is reduced to a upper triagular matri. Te te solutios are obtaied by back substitutio metod. 7. Eplai briefly Gauss Jorda metod. I Gauss Jorda metod, te coefficiet matri is reduced to idetity matri or a diagoal matri. Te te solutios are obtaied directly witout usig back substitutio metod. 8. Distiguis Gauss Elimiatio metod ad Gauss Jorda metod. Gauss Elimiatio metod (i Coefficiet matri of te give system is reduced to a upper triagular matri. (ii Back substitutio process gives solutio. Gauss Jorda metod Coefficiet matri of te give system is reduced to idetity matri or a diagoal matri. Solutios are obtaied directly. 9. Wat is meat by diagoally domiat system? If i eac equatio of te give system, te absolute value of te largest coefficiet is greater ta te sum of te absolute values of all te remaiig coefficiets te te system is said to be diagoally domiat. i.e., if A = ( for all i. 0. Solve + y = ; + 3y = 5 by Gauss Elimiatio metod. [A,B] = = R R R Te solutio is =, y =.

3 . Solve 3 + y = 4 ; - 3y = 7 by Gauss Jorda metod. [A,B] = = R 3R R = R R /-3 = R R R = R R /3 Te solutio is =, y =.. State te coditio for te covergece of Gauss Jacobi metod ad Gauss-Seidel metod. Gauss Jacobi metod ad Gauss-Seidel metod will coverge if te give system of equatios is diagoally domiat. 3. Compare Gauss Jacobi ad Gauss-Seidel metods for solvig liear systems of te form AX=B. Gauss Jacobi metod Gauss-Seidel metod (i Uses te value of i obtaied i previous step. (ii Slow covergece Uses te value of i obtaied i curret step. Fast covergece (Twice faster ta Jacobi metod 4. We te metod of iteratio will be useful? Metod of iteratio will be useful if te coefficiet matri of te give system of equatios is diagoally domiat. 6. We will te solutio of AX=B by Gauss-Seidel metod coverge quickly? Te solutio of AX=B by Gauss-Seidel metod will coverge quickly if te coefficiet matri is diagoally domiat. 7. Wat type of eigevalue ca be obtaied usig power metod? We ca obtai domiat eigevalue of te give matri usig power metod. 8. For wat type of matrices, Jacobi s metod ca be used to fid eige values ad eigevectors? Jacobi s metod ca be used to fid eige values ad eigevectors of symmetric matrices. UNIT-II - Iterpolatio ad Approimatio. Write Newto s forward differece formula.. Write Newto s backward differece formula. 3. We will we use Newto s forward iterpolatio formula? Newto s forward iterpolatio formula is used we iterpolatio is required ear te begiig of te table ad for etrapolatio at a sort distace from te iitial value We will we use Newto s backward iterpolatio formula? Newto s forward iterpolatio formula is used we iterpolatio is required ear te ed of te table ad for etrapolatio closer to te rigt of y. 5. Wat is forward differece operator? Forward differece operator is deoted by ad is defied as 5. Gauss-Seidel metod is better ta Gauss Jacobi metod. Wy? I Gauss-Seidel metod te latest values of ukow at eac stage of iteratio are used i te et stage of iteratio. Hece te covergece of te Gauss-Seidel metod is faster ta Gauss Jacobi metod. 6. Wat is backward differece operator? Backward differece operator is deoted by ad is defied as

4 7. Fid. = log (+ log = 8. Takig to be te iterval of differecig, fid ( e ( e ( e = (e e = (e e e = e ( e = ( e e = ( e (e e = ( e e (e = e (e 9. State Lagrage s formula for iterpolatio. ( ( ( 3...( y( y0 ( 0 ( 0 ( ( 0 ( 0 ( ( 3...( y ( 0 ( ( 3...( ( 0 ( (... ( y ( 0 ( (...( 0. State Iverse Lagrage s formula for iterpolatio. ( y y ( y y ( y y3...( y y ( y 0 ( y0 y ( y0 y( y0 y3...( y0 y ( y y0 ( y y ( y y3...( y y ( y y0 ( y y( y y3...( y y ( y y0 ( y y ( y y... ( y y ( y y0 ( y y ( y y...( y y ( ( y( y y ( ( Wic metod ca be used for bot equal ad uequal itervals? Lagrage s Metod ca be used for bot equal ad uequal itervals. 3. Give te divided differece iterpolatio formula f ( y ( y ( ( y ( ( ( y Wat is a cubic splie? A cubic polyomial wic as cotiuous slope ad curvature is called a cubic splie. 5. Wat is a atural cubic splie? A cubic splie fitted to te give data suc tat te ed cubics approac liearity at teir etremities is called a atural cubic splie. 6. State te coditios for a atural cubic splie. A cubic splie g( fits to eac of te poits is cotiuous ad is cotiuous i slope ad curvature suc tat ad is called a atural cubic splie. [assume tat (, I = 0,,,, are data poits]. 7. State cubic splie formula 3 3 y f ( [( i Mi ( i Mi] 6 ( i [ yi Mi ] 6 ( i [ yi Mi ] 6 i =,,., were i =,,,- wit.. Costruct a liear iterpolatig polyomial for te give poits ( 0, y 0, (, y usig Lagrage s formula.

5 UNIT-III- NUMERICAL DIFFERENTIATION AND INTEGRATION. We ca umerical differetiatio be used? We te fuctio is give i te form of table of values istead of givig aalytical epressio, we use umerical differetiatio.. State Newto s forward formula for fidig first ad secod derivatives. dy 3 [ y0 (u y0 (3u 6u y0 d! 3! ( u u u y 4! 0 ( u u u u y 0...] 5! d y 3 4 [ y 0 ( u y0 (6u 8u y0 d ( u u u y 0...] 3. State Newto s forward formula for fidig first ad secod derivatives at = State Newto s backward formula for fidig first ad secod derivatives at =. 6. Write Newto Cote s formula o itegratio. 7. Write Trapezoidal rule for itegratio. 8. Write Simpso s rule (or Simpso s rule for itegratio. 9. Write Simpso s rule for itegratio. 4. State Newto s backward formula for fidig first ad secod derivatives. dy 3 [ y (u y (3u 6u y d! 3! ( u u u y 4! ( u u u u y...] 5! d y 3 4 [ y ( (6 8 u y u u y d ( u u u y...] 0. Wic formula is called closed formula? Simpso s rule (or Simpso s rule is called closed formula.. Wat is te coditio to apply Simpso s rule for itegratio? Te iterval of itegratio must be divided ito a eve umber of subitervals.. Wat is te coditio to apply Simpso s rule for itegratio? Te iterval of itegratio must be divided ito a 3-multiple umber of sub-itervals. 3. We Simpso s rule ca be used?

6 Te iterval of itegratio is divided ito a eve umber of subitervals. 4. Wat is te order of error i Trapezoidal rule? Te order of error i Trapezoidal rule is. 5. Wat is te order of error i Simpso s rule (or Simpso s rule? Te order of error i Simpso s rule is 6. Wat is te order of error i Simpso s rule? Te order of error i Simpso s rule is 7. Wat is te error i Trapezoidal rule? Te error is 8. Wat is te error i Simpso s rule (or Simpso s rule? [ sum of te values of f at four corers + (sum of te values of f at te remaiig odes o te boudary + 4(sum of te values of f at te iterior odes] 4. State Simpso s rule for Double itegratio. [ sum of te values of f at four corers + (sum of te values of f at te odd positios o te boudary ecept corers + 4(sum of te values of f at te eve positios o te boudary ecept corers + {4(sum of te values of f at odd positios + 8(sum of te values of f at eve positios o te odd rows of te matri ecept boudary rows} + {8 (sum of te values of f at odd positios + 6 (sum of te values of f at eve positios o te eve rows of te matri ecept boudary rows} ] UNIT-IV - Iitial value problems for Ordiary Differetial Equatios Te error is 9. Wat is te error i Simpso s rule? Te error is 0. If I = ad I = wit = 0.5 ad = 0.5, fid I usig Romberg s metod.. State Gaussia -poit quadrature formula for itegratio.. State Gaussia 3-poit quadrature formula for itegratio.. Write dow Taylor series formula.. Give y = + y, y(0 =. Fid y(0.by Taylor series metod. 0 = 0 ad y 0 = y = + y y 0 = y = + y y 0 = y = y y 0 = y IV = y y IV 0 = = = y(0. = = State Trapezoidal rule for Double itegratio. 3. Wat are te merits ad demerits of Taylor series metod? Merits: It is a powerful sigle step metod. It is te metod if te epressio for iger order derivatives are simpler. Demerits: Te major demerit of tis metod is te evaluatio of iger order derivatives become tedious for complicated algebraic epressios.

7 4. State Euler s formula., = 0,,, 4. Fid y(0. by Euler s metod, give tat 9. Wic metod is better Taylor series metod or R-K metod? Wy? R-K metod is better sice iger order derivatives of y are ot required. Taylor series metod ivolves iger order derivatives wic may difficult i case of complicated algebraic fuctios. = 0.( = y y 0 0 Y? 5. Write modified Euler s formula. 6. State Ruge-kutta fourt order formula for solvig first order differetial equatios. were 7. State Ruge-kutta fourt order formula for solvig first order differetial equatios. ad were 0. Wat are te distiguised properties of R-K metod? (i Tis metod do ot require te iger order derivatives ad requires oly te fuctio values at differet poits. (ii To evaluate, we eed oly but ot previous y values.. Name te sigle step metods. Taylor series metod, Euler s metod, Ruge-Kutta metod are sigle step metods.. Name te multi-step metods. Adam s predictor corrector metod ad Mile s predictor-corrector metod are multi-step metods. 3. Write Adam s predictor corrector formula. Predictor formula : Corrector formula : 4. Write Mile s predictor corrector formula. Predictor formula : Corrector formula : 5. Wat is te error i Adam s metod? Predictor error : corrector error : 6. Write te fiite differece Beder-Scimidt eplicit sceme to solve te 8. Wat is te order of error i R-K metod fourt order formula? Te order of error i R-K metod fourt order formula is 5. oe dimesioal eat equatio

8 7. Write te fiite differece eplicit sceme to solve te oe dimesioal wave equatio 8. Classify te partial differetial equatio Here A =, B = 0, C= y B 4AC = -4y<0 Te give equatio is elliptic. Note : Geeral Equatio of a PDE is. i If B 4AC <0, te te give equatio is elliptic. ii If B 4AC =0, te te give equatio is parabolic. iii If B 4AC >0, te te give equatio is yperbolic. 9. Wat is te classificatio of Here A =, B =, C = B 4AC = 4 4 = 0 Te give equatio is parabolic. 0. Wat is te classificatio of Here A =, B = -3, C = 0 B 4AC = 9 0 > 0 Te give equatio is yperbolic.. Give te fiite differece sceme to solve te Laplace equatio umerically. (or Give te stadard five poit formula to solve Laplace s equatio.. Write te diagoal five poit formula to solve Laplace s equatio. 3. Wat is te purpose of Leibma s process? Te purpose of Leibma s process is to fid te solutio of Laplace equatio by iteratio over a square wit boudary values. UNIT-V Boudary value problems for Ordiary ad Partial Differetial Equatios. Write te fiite differece Beder-Scimidt eplicit sceme to solve te oe dimesioal eat equatio. Write te fiite differece Implicit Crak-Nicolso s sceme to solve te oe dimesioal eat equatio 3. Write te fiite differece eplicit sceme to solve te oe dimesioal wave equatio 4. Classify te partial differetial equatio Here A =, B = 0, C= y B 4AC = -4y<0 Te give equatio is elliptic. 5. Wat is te classificatio of Here A =, B =, C = B 4AC = 4 4 = 0 Te give equatio is parabolic. 6. Wat is te classificatio of Here A =, B = -3, C = 0 B 4AC = 9 0 > 0 Te give equatio is yperbolic. 7. Give te fiite differece sceme to solve te Laplace equatio umerically. (or Give te stadard five poit formula to solve Laplace s equatio.

9 8. Write te diagoal five poit formula to solve Laplace s equatio. 9. Wat is te purpose of Leibma s process? Te purpose of Leibma s process is to fid te solutio of Laplace equatio by iteratio over a square wit boudary values. 0. Give te fiite differece sceme to solve te Poissso equatio umerically.. Write te fiite differece formula for solvig ordiary differetial equatios.

The Advection-Diffusion equation!

The Advection-Diffusion equation! ttp://www.d.edu/~gtryggva/cf-course/! Te Advectio-iffusio equatio! Grétar Tryggvaso! Sprig 3! Navier-Stokes equatios! Summary! u t + u u x + v u y = P ρ x + µ u + u ρ y Hyperbolic part! u x + v y = Elliptic