Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this re is the sudden chnge in temperture. We re given the temperture vs. depth dt for lke in ble. ble emperture vs. depth for lke. emperture, C Depth, m 9. 9. 9 8.8 8.7 4 8. 5 8. 6 7.6 7.7 8 9.9 9 9. 5..
5.. Chpter 5. Figure emperture vs. depth of lke. Using the given dt, we see the lrgest chnge in temperture is between 8 m nd 7 m. Determine the vlue of the temperture t 7.5 m using the direct method of interpoltion nd first order polynomil. Solution For first order polynomil interpoltion (lso clled liner interpoltion), we choose the temperture given by y x, y f x x, y x
Direct Method of interpoltion More Exmples: Civil Engineering 5.. Figure Liner interpoltion. Since we wnt to find the temperture t 7.5 m, nd we re using first order polynomil, we need to choose the two dt points tht re closest to 7.5 m tht lso brcket 7.5 m to evlute it. he two points re 8 nd 7. hen,. gives 7 7. 6 8 7, 8 8. 7 7 7. 6 7 Writing the equtions in mtrix form, we hve 8.7 7 7.6 Solving the bove two equtions gives 58.9 nd 5. 9 Hence 58.9 5.9, 8 7 7.5 58.9 5.9 7.5 4.65C Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this re is the sudden chnge in temperture. We re given the temperture vs. depth dt for lke in ble. ble emperture vs. depth for lke. emperture, C Depth, m 9. 9. 9 8.8 8.7 4 8. 5 8. 6 7.6 7.7 8 9.9 9 9.
5..4 Chpter 5. Using the given dt, we see the lrgest chnge in temperture is between 8 m nd 7 m. Determine the vlue of the temperture t 7.5 m using the direct method of interpoltion nd second order polynomil. Find the bsolute reltive pproximte error for the second order polynomil pproximtion. Solution For second order polynomil interpoltion (lso clled qudrtic interpoltion), we choose the velocity given by v t t t y x, y x, y f x x, y x Figure Qudrtic interpoltion. Since we wnt to find the temperture t 7. 5, nd we re using second order polynomil, we need to choose the three dt points tht re closest to 7. 5 tht lso brcket 7. 5 to evlute it. he three points re 9, 8 nd 7. (Choosing the three points s 8, 7 nd 6 is eqully vlid.) hen, 9. gives 9. 7 7. 6 9 8, 7, 9 9 9 9. 9 8 8 8. 7 7 7 7 7. 6 Writing the three equtions in mtrix form
Direct Method of interpoltion More Exmples: Civil Engineering 5..5 9 8 9.9 8 64.7 7 49 7.6 nd the solution of the bove three equtions gives 7.7 6.65.5 Hence 7.7 6.65.5, 9 7 At 7. 5, 7.5 7.7 6.65 7.5.5 7.5 4.8C he bsolute reltive pproximte error obtined between the results from the first nd second order polynomil is 4.8 4.65 4.8.65% Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this re is the sudden chnge in temperture. We re given the temperture vs. depth dt for lke in ble. ble emperture vs. depth for lke. emperture, C Depth, m 9. 9. 9 8.8 8.7 4 8. 5 8. 6 7.6 7.7 8 9.9 9 9. Using the given dt, we see the lrgest chnge in temperture is between 7 m. 8 m nd
5..6 Chpter 5. ) Determine the vlue of the temperture t 7.5 m using the direct method of interpoltion nd third order polynomil. Find the bsolute reltive pproximte error for the third order polynomil pproximtion. d b) he position where the thermocline exists is given where. Using the d expression from prt (), wht is the vlue of the depth t which the thermocline exists? Solution ) For third order polynomil interpoltion (lso clled cubic interpoltion), we choose the temperture given by y x, y x, y f x x, y x, y x Figure 4 Cubic interpoltion. Since we wnt to find the temperture t 7. 5, nd we re using third order polynomil, we need to choose the four dt points closest to 7. 5 tht lso brcket 7.5 to evlute it. he four points re 9, 8, 7 nd 6. hen, 9. gives 9. 7 7. 6 8. 9 8, 7, 6, 9 9 9 9 9. 9
Direct Method of interpoltion More Exmples: Civil Engineering 5..7 8 8 8 8. 7 7 7 7 7 7. 6 6 6 6 6 8. Writing the four equtions in mtrix form, we hve 9 8 79 9.9 8 64 5.7 7 49 4 7.6 6 6 6 8. Solving the bove four equtions gives 65.9 6.58 5.55.5667 Hence 65.9 6.58 5.55.5667, 4.75C he bsolute reltive pproximte error nd third order polynomil is 4.75 4.8 4.75.9898% 9 6 7.5 65.9 6.58 7.5 5.55 7.5.5667 7. 5 obtined between the results from the second b) o find the position of the thermocline, we must find the points of inflection of the third d order polynomil, given by d ( ) 65.9 6.58 5.55.5667, 9 6 d 6.58 7. 4.7, 9 6 d d 7. 9.4, 9 6 d Simply setting this expression equl to ero, we get 7. 9.4 7.568 m his nswer cn be verified due to the fct tht it flls within the specified rnge of the third order polynomil nd it lso flls within the region of the gretest temperture chnge in the collected dt from the lke.
5..8 Chpter 5. INERPOLAION opic Direct Method of Interpoltion Summry Exmples of direct method of interpoltion. Mjor Civil Engineering Authors Autr Kw Dte November, 9 Web Site http://numericlmethods.eng.usf.edu