NUMERICAL METHODS FOR SOLVING EQUATIONS

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1 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

2 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page of 11 NUMERICAL METHODS FOR SOLVING EQUATIONS. May algebraic equatios caot be solved usig the stadard methods of factorisatio ad substitutio ito formulae, especially whe they are more complicated tha quadratics. Take the cubic equatio x - 4x + 6 = 0 This equatio caot be factorised, ad there is also o geeral formula for solvig cubic equatios as there is for quadratic equatios, so other methods must be used. Oe such method, studied i earlier years, is 'trial ad improvemet' usig a decimal search. This ivolves fidig a value of x which makes x - 4x + 6 as close to 0 as the accuracy allows. This method is o loger part of the GCSE syllabus, for reasos which will become clear. Example (1): ( Legacy example) The graph of y = x - 4x + 6 is show below, ad has three solutios of x - 4x + 6 = 0, give that its graph cuts the x-axis at three poits, with those solutios beig the x-coordiates of those poits.. Oe of the solutios is close to -1, aother is betwee 1 ad, ad a third oe is betwee ad 4. Use trial ad improvemet to fid the solutio betwee 1 ad correct to oe decimal place.

3 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page of 11 We kow that the solutio lies betwee 1 ad, so we begi with those trial values of x. Trial x Computed value of x - 4x + 6 Commet 1 (1) - 4(1) + 6 = Too high (> 0) () - 4() + 6 = - Too low (< 0) The solutio seems to be about half-way betwee 1 ad, so we ca substitute x = 1., 1.4, 1.5, ito the equatio. It must also be oted that x - 4x + 6 decreases as x icreases withi this iterval. Trial x Computed value of x - 4x + 6 Commet 1. (1.) - 4(1.) + 6 = 1.47 Too high ( > 0) 1.4 (1.4) - 4(1.4) + 6 = Too high 1.5 (1.5) - 4(1.5) + 6 = 0.75 Still too high, but gettig earer 1.6 (1.6) - 4(1.6) + 6 = Too low ( < 0) We ow kow that the solutio lies betwee 1.5 ad 1.6, so we substitute x = 1.55 (the mea of 1.5 ad 1.6 ) to obtai the correct first decimal place. Whe x = 1.55, x - 4x + 6 = 0.114, which is greater tha 0. The chage betwee the value of x - 4x + 6 from beig too high to beig too low occurs i the iterval from 1.55 to 1.6. Sice all values i this particular iterval roud to 1.6 to 1 decimal place, the required solutio of the equatio x - 4x + 6 = 0 is x = 1.6 to 1 decimal place.

4 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 4 of 11 Iterative methods. The decimal searchig techique used the first two examples was simple, but there was a drawback. The process was slow, requirig a large umber of trials for eve a moderate level of accuracy. Example (): Fid the solutio of x + 6x = 15 to four decimal places. The equatio x + 6x = 15 ca be rewritte as x + 6x 15 = 0, ad it has oe solutio betwee 1 ad, where the graph itersects the x-axis. Trial ad improvemet would be usuitable here, because of the large umber of trials eeded. If it takes o average five trials to achieve accuracy to oe decimal place, it would take four times as log, or about 0 attempts, to achieve four decimal-place accuracy. This calls for a differet method kow as iteratio. I iteratio, we use a iitial estimated solutio as the iput to a formula, ad the resultig output provides a improved estimate. This improved estimate ca the be iput ito the formula agai, util the desired level of accuracy is reached.

5 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 5 of 11 The equatio x + 6x = 15 has a sigle solutio betwee 1 ad. Use the followig flowchart to calculate this solutio to four decimal places. Choose a startig value of x =, ad keep four decimal places i the workig. You may otice that the expressio x x 15 6 does ot have ay obvious relatioship with the equatio x + 6x = 15, but fidig such iterative formulae is outside the scope of GCSE. All questios will have a suitable formula provided for you!

6 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 6 of 11 ( ) 15 ( ) 6 By choosig the startig value of x = as iput, we calculate The ew value of x does ot agree sufficietly with the iput value of, so we take the No brach ad update x with the ew value of 1.7 ito the formula to obtai (1.7 ) (1.7 ) 6 brach agai This result is still ot close eough to 1.7, so we take the No (1.696 ) 15 (1.696 ) 6 Substitutig x = gives us the ext value of x, amely The iteratios appear to be closig i o the required solutio, with oly the fourth decimal place differet after this last calculatio. We still take the No brach, substitute x = 1.69, givig our ext value of x which is (1.69 ) (1.69 ) 6 to 4 decimal places. Sice this output value of x ow agrees to four decimal places with the iput, we ca ow ed the iterative loop ad take the Yes brach. The output value of x = 1.69 is the solutio of the equatio x + 6x = 15 to the guarateed accuracy of four decimal places after four iteratios. This method is more accurate tha the decimal search, ad is also a great deal quicker. We ca check by substitutig x = 1.69 ito x + 6x; the result is , very close to 15. Aother way of expressig a iterative formula without a flowchart is to use subscripts where x 1 is the startig trial value of the solutio ad x, x... are the subsequet iteratios. This is similar to geeratig sequeces by iductive defiitio see Sequeces documet. The formula used i the last example could be defied as where x is the curret estimate for x ad x +1 is the ext oe. x 1 x x 15 6 Thus i the last example, we had x 1 =, x = 1.7, x = ad x 4 = 1.69.

Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 7 of 11 A very useful hit.

This example has the As key o the bottom row. For the last example, we ca set the startig value by pressig the keys = =.

) Next, we eter the iterative formula usig the As butto, amog others.

7 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 7 of 11 A very useful hit. I the last example, we had bee calculatig iterated values of x, roudig them to four decimal places, ad re-iputtig the ew values ito the iterative formula. The work was rather tedious, but fortuately, most calculators have a As key to perform such calculatios very rapidly. This example has the As key o the bottom row. For the last example, we ca set the startig value by pressig the keys = =. ( O more moder calculators, the = key ca be pressed oce oly.) Next, we eter the iterative formula usig the As butto, amog others. (We must eter As followed by x whe eterig the top lie, similarly As followed by x whe eterig the bottom lie). Havig started with the value of x 1 =, it is just a matter of repeatedly pressig the = key to obtai the values of x, x ad so forth. Notice how the value of x appears as a fractio, ad how rapidly the values coverge to 4-decimal place accuracy.

8 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 8 of 11 Example (): The equatio x - 4x + 6 = 0 from Example (1) had a solutio of 1.6 to oe decimal place. Use the followig flowchart to calculate this same solutio to four decimal places, ad practise by usig the As key o the calculator. Choose a startig value of x =.

9 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 9 of 11 By choosig the startig value of x 1 = as iput, we calculate. x x O the calculator, it is a matter of pressig the, =, ad = keys, followed by eterig As ( As), ad the pressig the = key throughout. This takes care of all the calculatios, iasmuch as we just press the = key util sufficiet accuracy has bee reached. I other words, a No result meas press the = key agai, ad a Yes meas Ed of sum. Pressig the = key first time gives us the ext trial value of x, amely x, or 1.75, which does ot agree sufficietly with x 1, so we press the = key agai to give us x, or Pressig the = key agai gives us x 4 = 1.571, x 5 = ad x 6 = Sice the values of x 5 ad x 6 agree to four decimal places, we ow take the Yes brach ad ed the iterative process.. Oe solutio of the equatio x - 4x + 6 = 0 is to 4 decimal places.

10 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 10 of 11 This example could have bee rephrased as follows, without the eed of a flowchart : Example () Versio : The equatio x - 4x + 6 = 0 has a solutio betwee 1 ad. x x 1 ad 1 x Use the iterative formula x, x, x 4 ad x 5 to four decimal places. I tabular form : x x to fid the values of Example (b) : Repeat the last example, usig the iterative formula x x 1 1. x (8 x ) O calculator: press the, =, ad = keys, eter throughout. As 1 As(8 As), the keep pressig the = key Results i tabular form : x

11 Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 11 of 11 Example (4): The equatio x - 4x + 6 = 0 also has a solutio betwee -1 ad -. Use the iterative formula x, 0.6 x 9 x, x 4 ad x 5 to four decimal places x ad 1 1 x x to fid the values of 0.6 As Calculator: press the -1, =, ad = keys, eter ( As), ad press the = key throughout. I tabular form : x Example (5): The equatio x - 4x + 6 = 0 also has a solutio betwee ad 4. i) Use the iterative formula x, x 1 5x 11x x, x 4 ad x 5 to four decimal places. 6 ad x 4 to fid the values of 4x 1 Calculator: press the 4, =, ad = keys, eter 5As 6 11As 4As, the keep pressig the = key.. i) I tabular form : x

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as