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The Daniels/Gilbert Formula--Page 8

Here we are going to solve the race prediction equation using the Newton-Raphson method. In the beginning we are going to use a VDOT (oxygen uptake) score of 83 ml/kg of body weight/min and fix the distance at 5000 meters. I have built these figures into the Javascript code as fixed constants. I have also built-in a condition that tells the Javascript loop to stop looping when the time guessed equals the same time guessed during the previous iteration and rounded off to the 6th decimal point.

We are using here the same oxygen uptake I used in the interactive example on Page 4. I had used a curve-fit equation on that page. Let's see how the Newton-Raphson method comes to the same conclusion. I am using a first guess of 18. You can experiment substituting other guesses in space provided if you like. (Click the iteration below to converge on the correct value of t.)

f(t)= ((((0.000104) * (pow (5000, 2)) * (pow (t, -2))) + ((0.182258)*5000*(pow (t, -1))) -4.6)/((0.2989558*exp( -0.1932605*t)) + (0.1894393 * exp(-0.012778*t)) + 0.8)) - 83, f'(t)= ((((0.2989558*exp( -0.1932605*t)) + (0.1894393 * exp(-0.012778*t)) + 0.8)*((-0.000208)*(pow(5000, 2)) * (pow(t,-3))) - ((0.182258) * 5000 * (pow (t, -2)))) - (81.8 * ((0.2989558)*(exp( -0.1932605*t)) + (0.1894393) * (exp(-0.012778*t))))) / pow (((0.2989558*exp( -0.1932605*t)) + (0.1894393 * exp(-0.012778*t)) + 0.8), 2)
Enter time guess t =
Notice above that the values of f(t) and f'(t) are much more complex in nature than the example (f(t)=4*t-8 with f'(t)=4) I gave you on the previous page to illustrate Newton-Raphson. Because the prediction equation is so much more complex, it took more iterations to solve for the true value of t.

When you use an initial guess of 18 minutes for the predicted time, Newton-Raphson kicks out its answer rounded to the 6th decimal place of 12.883892 (12:53) after only 39 iterations. I have given the Javascript a looping capability of as many as 100 iterations in which to find the answer, but it did not need that many iterations in this case.

Try substituing other guesses and see how many iterations it takes to hit the same predicted "drop dead" time performance. You can even guess ridiculously with 1 minute or as much as 366 minutes and it will still kick out the correct answer. It goes to show that guy Newton was pretty smart--his math not only computes your best performance time, but he is also the guy who came up with the equations of how hard and how fast you hit the pavement when you trip over the finish line!

A word of caution: As I mentioned earlier, oxygen uptake alone is not a good predicter of race performance capability because there are other factors such as how soon a runner will hit his or her lactate turnpoint that mitigate its effective influence on performance either positively or negatively. Daniels and Gilbert, with their "drop dead" equation forced every runner into the same running economy category as that of an elite athlete. Few people have that type of profile in reality.

So in the next installment, I will show you how you can treat the VO2max value in the equation more like a composite index by first telling the Javascript program your current best measured performance at a given distance. In that manner, the pedictions will become very accurate because factors like running economy (how soon you reach lactate turnpoint) are a composite part of your PR. The runner will be thus assigned in the algorithm an "effective VO2max" score based on their actual performance.