How to Use the Newton Raphson Method of Quickly Finding Roots

Step-by-Step Guide

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   Step 1: Format cell B1 of a new worksheet with a thick outline and color canary yellow -- this means that it is an input cell.

   In cell A1, type, "Square" (w/o the quotation marks) and adjust column width to fit the phrase.
   
   Select column B and Format Cells Number for 14 decimal places.
   
   Expand the column width if need be. , In this example, it is
   685. , In cell A2, type "InitialROOTGuess" (w/o the quotation marks).
   
   Since 25^2 = 625, and 30^2 = 900, we will use
   25.
   
   Please input 25 in cell B2 , Doing so will assign the defined variable name "Square" to cell B1, as well as the defined variable name "InitialROOTGuess" to cell B2.
   
   The Newton-Raphson method is that guess X(1) (read X sub 1) = X(0) + f(X(0))/f'(X(0)).
   
   In this case, where x^2 = 685, f(x) = x^2
   - 685 = 0 and f'(x) = 2x.
   
   Therefore, into cell B3, please input the following formula (w/o quotation marks) "=InitialROOTGuess-((InitialROOTGuess^2-Square)/(2*InitialROOTGuess))"
   
   The formula will be taking each previous guess and using it to get a better current answer by a factor of 2 more significant digits each computation.
   
   You should see the correct answer,
   26.1725046566048, which is the square root of 685, in cell B6.
   
   Please notice how fast the Newton-Raphson method arrived at the correct answer. , For simple expressions such as 4x^3, to obtain the derivative, one takes the exponent down and multiplies it as a (combined) coefficient, and reduces the exponent by 1 power, so the derivative of x^n = nx^(n-1) and the derivative of 4x^3 = 12x^2.
   
   If there is a constant such as π in the expression as a factor, that is treated as the 4 just was above in 4x^3
   -- it remains.
   
   If there is a lone x, such as in the expression 4x^3
   - x, the derivative of that is 12x^2
   - 1, since x^0 =
   1. if there is a constant added such as 4x^3 + 10, it is dropped, and the derivative is 12x^2. ,, In the new cell A1, type "Power" (w/o quotation marks).
   
   Format cell C1 with a border and red font and input 10 into cell C1. , when Power =
   10.
   
   Therefore, we have (a+10) = a^10, so F(X) = a^10
   - a
   - 10 and F'(x) = 10*a^9
   -
   1.
   
   Our InitialROOTGuess will be 1 since 1+10 = 11 and 1^10 = 1, but 2+10 = 12 and 2^10 = 1024 (which is much too large).
   
   We do not know the "square"
   
   which is actually a^10
   -- not a square
   -- or sum (a+b) yet.
   
   Do not input an InitialROOTGuess yet however.
   
   Let's preserve the square root function we made in columns A and B by moving over and working in column C. , Select cell C1 and Insert Name Define Powerful for it.
   
   You may type "Powerful" into B1 if you so choose.
   
   Select cell C3 and input the following formula (w/o quotes): "=InitialGuess+(((InitialGuess)^Powerful)-Powerful-InitialGuess)/(Powerful*(-InitialGuess)^(Powerful-1)".
   
   The evaluated result should equal
   2.
   
   This has modified the derivative in the denominator somewhat by dropping the constant; in this case, that is necessary, along with changing the sign of the InitialGuess in the final term.
   
   Call it the Garthwaite method if you please
   -- it is part of trying to arrive at the base instead of the root. ,, An InitialGuess of
   -1 will result in the answer
   -1.24233531640777 and that is also correct in that (-1.24233531640777 + 10) =
   -1.24233531640777^10 =
   8.75766468359223 , You're done! ,,,
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   Step 2: Input the number of which the root is to be found in cell B1.

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   Step 3: In cell A1

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   Step 4: type "Square".

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   Step 5: Select cell range A1:B2 and Insert Name Create in Left Column.

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   Step 6: And now select cell range B4:B15 and input the following formula

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   Step 7: "=B3-((B3^2-Square)/(2*B3))" and Edit Fill Down so that the formula calculates all the way down to cell B15 for large numbers.

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   Step 8: Bear in mind that this method can find roots for complicated equations like x^5 * (4x^2 +1)^2 so long as one can take the first derivative (see Calculus I and II).

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   Step 9: Find the answer to the Neutral Operation of (a+b) = a^b

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    Step 10: called "neutral" between the operations of Addition and Exponentiation on either side of the equals sign.

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    Step 11: Select Row 1 and Insert Row.

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    Step 12: Try to find (a+b) = a^b when b=10

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    Step 13: Select cell C2 and Insert Define Name InitialGuess for it and you may type "InitialGuess" into cell B2 if you so choose.

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    Step 14: Select cell range C4:C15 and input into C4 the formula "=C3+(((C3)^Powerful)-Powerful-C3)/(Powerful*(-C3)^(Powerful-1))" and Edit Fill Down so that cell C15 contains the computed result of 1.27411386474511

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    Step 15: which is the answer we seek.

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    Step 16: (1.27411386474511 + 10) = 1.27411386474511^10 = 11.2741138647451 and the problem of (a+b) = a^b when b = 10 has been solved.

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    Step 17: Save the workbook.

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    Step 18: Final image:

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    Step 19: See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel

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Detailed Guide

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