Depending on your needs, a simple divide-and-conquer strategy can be used. It won't converge as fast as some other methods but it may be a lot easier for a novice to understand. In addition, since it's an O(log n) algorithm (halving the search space each iteration), the worst case for a 32-bit float will be 32 iterations.
Let's say you want the square root of 62.104. You pick a value halfway between 0 and that, and square it. If the square is higher than your number, you need to concentrate on numbers less than the midpoint. If it's too low, concentrate on those higher.
With real math, you could keep dividing the search space in two forever (if it doesn't have a rational square root). In reality, computers will eventually run out of precision and you'll have your approximation. The following C program illustrates the point:
#include <stdio.h>
#include <stdlib.h>
int main (int argc, char *argv[]) {
float val, low, high, mid, oldmid, midsqr;
int step = 0;
// Get argument, force to non-negative.
if (argc < 2) {
printf ("Usage: sqrt <number>\n");
return 1;
}
val = fabs (atof (argv[1]));
// Set initial bounds and print heading.
low = 0;
high = mid = val;
oldmid = -1;
printf ("%4s %10s %10s %10s %10s %10s %s\n",
"Step", "Number", "Low", "High", "Mid", "Square", "Result");
// Keep going until accurate enough.
while (fabs(oldmid - mid) >= 0.00001) {
oldmid = mid;
// Get midpoint and see if we need lower or higher.
mid = (high + low) / 2;
midsqr = mid * mid;
printf ("%4d %10.4f %10.4f %10.4f %10.4f %10.4f ",
++step, val, low, high, mid, midsqr);
if (mid * mid > val) {
high = mid;
printf ("- too high\n");
} else {
low = mid;
printf ("- too low\n");
}
}
// Desired accuracy reached, print it.
printf ("sqrt(%.4f) = %.4f\n", val, mid);
return 0;
}
Here's a couple of runs so you hopefully get an idea how it works. For 77:
pax> sqrt 77
Step Number Low High Mid Square Result
1 77.0000 0.0000 77.0000 38.5000 1482.2500 - too high
2 77.0000 0.0000 38.5000 19.2500 370.5625 - too high
3 77.0000 0.0000 19.2500 9.6250 92.6406 - too high
4 77.0000 0.0000 9.6250 4.8125 23.1602 - too low
5 77.0000 4.8125 9.6250 7.2188 52.1104 - too low
6 77.0000 7.2188 9.6250 8.4219 70.9280 - too low
7 77.0000 8.4219 9.6250 9.0234 81.4224 - too high
8 77.0000 8.4219 9.0234 8.7227 76.0847 - too low
9 77.0000 8.7227 9.0234 8.8730 78.7310 - too high
10 77.0000 8.7227 8.8730 8.7979 77.4022 - too high
11 77.0000 8.7227 8.7979 8.7603 76.7421 - too low
12 77.0000 8.7603 8.7979 8.7791 77.0718 - too high
13 77.0000 8.7603 8.7791 8.7697 76.9068 - too low
14 77.0000 8.7697 8.7791 8.7744 76.9893 - too low
15 77.0000 8.7744 8.7791 8.7767 77.0305 - too high
16 77.0000 8.7744 8.7767 8.7755 77.0099 - too high
17 77.0000 8.7744 8.7755 8.7749 76.9996 - too low
18 77.0000 8.7749 8.7755 8.7752 77.0047 - too high
19 77.0000 8.7749 8.7752 8.7751 77.0022 - too high
20 77.0000 8.7749 8.7751 8.7750 77.0009 - too high
21 77.0000 8.7749 8.7750 8.7750 77.0002 - too high
22 77.0000 8.7749 8.7750 8.7750 76.9999 - too low
23 77.0000 8.7750 8.7750 8.7750 77.0000 - too low
sqrt(77.0000) = 8.7750
For 62.104:
pax> sqrt 62.104
Step Number Low High Mid Square Result
1 62.1040 0.0000 62.1040 31.0520 964.2267 - too high
2 62.1040 0.0000 31.0520 15.5260 241.0567 - too high
3 62.1040 0.0000 15.5260 7.7630 60.2642 - too low
4 62.1040 7.7630 15.5260 11.6445 135.5944 - too high
5 62.1040 7.7630 11.6445 9.7037 94.1628 - too high
6 62.1040 7.7630 9.7037 8.7334 76.2718 - too high
7 62.1040 7.7630 8.7334 8.2482 68.0326 - too high
8 62.1040 7.7630 8.2482 8.0056 64.0895 - too high
9 62.1040 7.7630 8.0056 7.8843 62.1621 - too high
10 62.1040 7.7630 7.8843 7.8236 61.2095 - too low
11 62.1040 7.8236 7.8843 7.8540 61.6849 - too low
12 62.1040 7.8540 7.8843 7.8691 61.9233 - too low
13 62.1040 7.8691 7.8843 7.8767 62.0426 - too low
14 62.1040 7.8767 7.8843 7.8805 62.1024 - too low
15 62.1040 7.8805 7.8843 7.8824 62.1323 - too high
16 62.1040 7.8805 7.8824 7.8815 62.1173 - too high
17 62.1040 7.8805 7.8815 7.8810 62.1098 - too high
18 62.1040 7.8805 7.8810 7.8807 62.1061 - too high
19 62.1040 7.8805 7.8807 7.8806 62.1042 - too high
20 62.1040 7.8805 7.8806 7.8806 62.1033 - too low
21 62.1040 7.8806 7.8806 7.8806 62.1038 - too low
22 62.1040 7.8806 7.8806 7.8806 62.1040 - too high
23 62.1040 7.8806 7.8806 7.8806 62.1039 - too high
sqrt(62.1040) = 7.8806
For 49:
pax> sqrt 49
Step Number Low High Mid Square Result
1 49.0000 0.0000 49.0000 24.5000 600.2500 - too high
2 49.0000 0.0000 24.5000 12.2500 150.0625 - too high
3 49.0000 0.0000 12.2500 6.1250 37.5156 - too low
4 49.0000 6.1250 12.2500 9.1875 84.4102 - too high
5 49.0000 6.1250 9.1875 7.6562 58.6182 - too high
6 49.0000 6.1250 7.6562 6.8906 47.4807 - too low
7 49.0000 6.8906 7.6562 7.2734 52.9029 - too high
8 49.0000 6.8906 7.2734 7.0820 50.1552 - too high
9 49.0000 6.8906 7.0820 6.9863 48.8088 - too low
10 49.0000 6.9863 7.0820 7.0342 49.4797 - too high
11 49.0000 6.9863 7.0342 7.0103 49.1437 - too high
12 49.0000 6.9863 7.0103 6.9983 48.9761 - too low
13 49.0000 6.9983 7.0103 7.0043 49.0598 - too high
14 49.0000 6.9983 7.0043 7.0013 49.0179 - too high
15 49.0000 6.9983 7.0013 6.9998 48.9970 - too low
16 49.0000 6.9998 7.0013 7.0005 49.0075 - too high
17 49.0000 6.9998 7.0005 7.0002 49.0022 - too high
18 49.0000 6.9998 7.0002 7.0000 48.9996 - too low
19 49.0000 7.0000 7.0002 7.0001 49.0009 - too high
20 49.0000 7.0000 7.0001 7.0000 49.0003 - too high
21 49.0000 7.0000 7.0000 7.0000 49.0000 - too low
22 49.0000 7.0000 7.0000 7.0000 49.0001 - too high
23 49.0000 7.0000 7.0000 7.0000 49.0000 - too high
sqrt(49.0000) = 7.0000
find where [it](http://nlindblad.org/2007/04/04/write-your-own-square-root-function/) has been moved
as a lame excuse, let me plug the web archive. – Inference