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# Linear Algebra MATH 2270

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This 6 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 2270 at University of Utah taught by Nicholas Korevaar in Fall. Since its upload, it has received 35 views. For similar materials see /class/229924/math-2270-university-of-utah in Mathematics (M) at University of Utah.

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Date Created: 10/26/15

Complex eigenvalueseigenvectors example REVISITED Math 22702 Monday Nov 26 2001 Glucose insulin model see discussion on page 339 of the text Let Gt be the excess glucose concentration mg ofG per 100 ml ofblood say in someone s blood at time t hours Excess means we are keeping track of the difference between current and equilibrium quotfastingquot concentrations Similarly Let Ht be the excess insulin concentration at time t When blood levels of glucose rise say as food is digested the pancreas reacts by secreting insulin in order to utilize the glucose Researchers have developed mathematical models for the glucose regulatory system Here is a simpli ed linearized version of one such models It would be meant to apply between meals when no additional glucose is being added to the system gt restart withlinalg with plots Gt1aGt bHt Ht1cGtdHt A particular choice of constants could lead to a model for you For example Gt1 9 4Gt Ht1 1 9 Ht On November 16 we solved the initial value problem say right after a big meal when G0 i i 100 HO 0 Of course we know that for gt Amatrix22 9 4 l 9 9 4 A 1 9 The solution at time t is just given by A t times the initial vector We made a table of values by uncommenting the appropriate line in the doloop below gt vvector 100 0 for i from 1 to 30 do G i evalmAAiampv l H i evalmAAiampv 2 print iGi H i 0d gt We used complex arithmetic and evals and evects as well as Euler s formula to solve the IVP Here was our data gt eigenvects1 9 2000000000 1 1 2000000000 0 I 0 11 9 2000000000 1 1 2000000000 0 I 0 11 gt From which we deduced a closed form solution for G and H 39gt rsqrt85 thetaarctan29 Gtt gtlOOrAtcosthetat Htt gt50r tsinthetat r9219544457 92186689459 Gt t gt 100 rt cos9 t Ht t gt50 r sin9t 39gt soltnplotGttHtttOlOOcolorblack pictl fieldplot 1G 4H 1G 1H G 4O lOOH 15 40 actualpointplotseqGiHiil30 displayactualpictlsoltntitle numerical and analytic solutions nu e ca and anawhcsouhons 4QL39r39 fr r v v K I l l h l 1 1 N I I I 1307 I zar 1 1 z z f h l I 1 I 1 l 039 r l l I z r 39 I I v N x I 167 x x x x x t x n u w x t 40 20 l 40 5 Ox 1Q0 t x I y 1 I I I 1 1 I I 1 u u 1 I 1 t 39l NEWTODAY We t this special example into the general framework of page 34 of our class notes for Nov 26 gt 9 l q2 phiarctan209 gt circlpointplotseq50sinkphi 50coskphi kO30 circ2implicitplotx 2yA22500x 100lOO y lOOlOOcolorblack elliplpointplotseqlOOcoskphi50sinkphi kO30 ellip2implicitplotxA2lOOA2yA250A2lx lOOlOO y lOOlOOcolorblack displaycirclcirc2elliplellip2actualsoltntitle general decomposition picture Complex eigenvalueseigenvectors example Math 22702 Spring 2004 Adapted from N Korevaar Glucose insulin model see discussion on page 339 of the text Let Gt be the excess glucose concentration mg ofG per 100 ml ofblood say in someone s blood at time t hours Excess means we are keeping track of the difference between current and equilibrium quotfastingquot concentrations Similarly Let Ht be the excess insulin concentration at time t When blood levels of glucose rise say as food is digested the pancreas reacts by secreting insulin in order to utilize the glucose Researchers have developed mathematical models for the glucose regulatory system Here is a simpli ed linearized version of one such models It would be meant to apply between meals when no additional glucose is being added to the system gt restart with linalg with plots Warning the protected names norm and trace have been redefined and unprotected Warning the name changecoords has been redefined Gt1aGt bHt Ht1cGtdHt A particular choice of constants could lead to a model for you For example Gt1 09 04 Gt Ht 1 01 09 Ht Using techniques developed in class you can solve the initial value problem say right after a big meal when mm 100 H0 0 Of course we know that for gt Amatrix22 9 4 l 9 09 04 A 7 01 09 the solution at time t is just given by AAt times the initial vector We can make a table of values using the following loop 39 gt vvector 100 0 for i from 1 to 30 do G i evalmAAiampv l H i evalmAAiampv 2 printiGi Hi 0d 1 900 100 2 7700 1800 3 62100 23900 4 463300 277200 5 3060900 2958100 6 15715700 29683800 7 22706100 282869900 8 927124700 2568535200 9 1861826310 2218969210 10 2563231363 1810889658 11 3031264090 1373477556 12 3277528704 9330033912 13 3322977189 5119501819 14 3195459544 1284574447 15 2927296567 2039342541 16 2552993208 4762704854 17 2107185693 6839427577 18 1622890021 8262670512 19 1130094198 9059293481 20 6547130390 9283458332 21 2179034020 9009825538 22 1642799596 8326746386 23 4809218190 7329791787 24 7260213086 6115890789 25 8980548094 4778280402 26 9993805445 3402397552 27 1035538392 2062777252 28 1014495643 08209611340 29 9458845240 02756306220 30 8402708470 1193952084 We can use complex eigenvalues and eigenvectors to solve the system Here is our data gt eigenvects1 09 02000000000 I 1 2000000000 0 I 011 09 02000000000 I 1 2000000000 0 I 0 11 E gt From which we can deduce a closed form solution for G and H gt rsqrt85 thetaarctan 2 9 Gtt gtlOOrAtcosthetat Htt gt50r tsinthetat r 09219544457 0 02186689459 Gt t gt 100 r cos0 t Ht r gt 50 r sin9 t 39gt soltnplotGttHtttOlOOcolorblack pictlfieldplot lG 4H lG lH G40 lOOH15 40 actualpointplotseqGiHiil30 displayactualpictlsoltntitle numerical and analytic solutions nulmeric and anayhcsouhons 4ggtrzKgrvivvvv N f 367 1 z z 1 z 1 x 1 I 1 f F N x 1 39 K 1 I 7 l k v x x x x x x x x x K x x x t w k K R 4o 20 40 60 gm 160 x v x x N v v x N N y 1 y w w x x I y r v 1 1 1 W W

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