WEBVTT 00:00.120 --> 00:03.930 Hello again! In this video, we are going to look at further numeric algorithms. 00:04.860 --> 00:10.350 We have already looked at some of the algorithms in the numeric header, like iota(), and generate(). The 00:10.350 --> 00:15.210 ones in this video are really going to be of interest, mainly, to people who do numeric computing. 00:15.870 --> 00:21.330 And if you are one of these people, you might be interested in C++ 17, which provides a number of 00:21.390 --> 00:22.410 special functions. 00:23.040 --> 00:27.540 So things like Bessel functions, Legendre polynomials, elliptic integral and so on. 00:28.440 --> 00:29.500 Those are in the header. 00:30.150 --> 00:31.830 I will leave you to investigate those on your own. 00:32.070 --> 00:34.080 It's a bit too specialized for this course, 00:34.080 --> 00:34.440 really! 00:36.420 --> 00:39.240 The first algorithm we are going to look at is partial_sum(). 00:39.720 --> 00:44.990 This will take an iterator range, and it will go through the iterator range, and it will calculate the running 00:45.000 --> 00:46.140 total of the elements. 00:46.890 --> 00:53.430 So the first element has a running total of 'a', the second test has a running total of a + b, 00:53.430 --> 00:54.510 a + b + c, and so on. 00:55.050 --> 01:00.510 And it will write each of these running totals to the corresponding element in a destination. 01:03.900 --> 01:08.910 So if we have a vector with elements 1, 2, 3, 4, 5, the elements in the destination 01:08.910 --> 01:12.390 will be 1, 1 + 2, 1 + 2 + 3 and so on. 01:13.620 --> 01:17.190 And this is in fact, how you can perform numerical integration. 01:17.730 --> 01:23.070 If this vector contains the value of a function at some points, then this vector will contain 01:23.070 --> 01:26.160 the values of the integral of that function at the same points. 01:28.970 --> 01:30.500 Okay, so let's try this out. 01:30.830 --> 01:33.740 So we have our vector, with elements 1, 2, 3, 4, 5. 01:34.310 --> 01:38.270 We have our destination. And we have our call to partial_sum(). 01:39.650 --> 01:40.460 Let's see what we get. 01:42.830 --> 01:49.570 Okay, so 1 is 1, 1 + 2 is 3, 1 + 2 + 3 is 6, and so on. 01:55.460 --> 01:57.680 The next algorithm adjacent_difference(). 01:58.010 --> 02:02.540 This will go through the iterator range, and this time it will write the difference between each element. 02:03.170 --> 02:09.200 So 'a' minus nothing is 'a', b - a and c - b, and so on. 02:09.770 --> 02:12.500 And these differences will get written to the destination. 02:12.770 --> 02:17.300 So the destination has elements a, b - a, c - b, and so on. 02:19.430 --> 02:25.550 If we give the results of partial_sum(), so this is the vector we got when we ran the partial_sum() 02:25.880 --> 02:31.940 calculation. If we give this as the argument to adjacent difference, then 3 - 1 is 02:31.940 --> 02:33.980 2, 6 - 3 is 3, and so on. 02:34.400 --> 02:35.510 And we get back the original 02:35.510 --> 02:35.990 vector. 02:36.590 --> 02:43.700 So adjacent_difference() is the inverse of partial_sum(), and the inverse of numeric integration is numeric 02:43.700 --> 02:44.510 differentiation. 02:49.230 --> 02:51.480 So let's see if we get the original vector back. 02:51.510 --> 02:57.000 So we have the results of the partial_sum() and then we pass that to adjacent_difference(). 03:00.100 --> 03:02.290 And there we are, we do get back the original vector. 03:07.600 --> 03:14.020 Finally, there is inner_product(). This takes 2 iterator ranges. It goes through both ranges 03:14.020 --> 03:19.930 and it multiplies together the corresponding elements, so a1 * b1, a2 * b2 and 03:19.930 --> 03:20.350 so on. 03:20.920 --> 03:26.320 And then it adds them all up. And that will calculate the scalar product of two vectors. 03:28.940 --> 03:33.650 So this is quite similar to the accumulate() algorithm, if you remember that, except we are doing a 03:33.650 --> 03:39.920 preliminary step where we multiply together the elements. When we call the accumulate() algorithm, we 03:39.920 --> 03:45.140 have to provide an initial value for the sum and we need to do that for inner_product() as well. 03:46.520 --> 03:51.980 So we pass it the two iterator ranges, and the return value will be the results of multiplying 03:51.980 --> 03:57.740 together pairs of elements from these iterators, and adding them together, and adding on this initial 03:57.740 --> 03:58.100 value. 04:02.300 --> 04:05.900 So let's try this out. We have the two vectors we have used so far. 04:07.430 --> 04:10.670 We passed these as the iterator ranges for inner_product(). 04:11.510 --> 04:18.020 We have the initial range of the sum as zero. And then the result of this will be the result of 04:18.020 --> 04:21.680 1 * 1 plus 3 * 2 plus 6 * 3, and so on. 04:23.420 --> 04:25.520 And, as I am sure you all know, that is 140! 04:27.710 --> 04:29.420 Okay, so that's it for this video. 04:29.780 --> 04:30.620 I will see you next time. 04:30.860 --> 04:32.660 Until then, keep coding!