WEBVTT 00:00.060 --> 00:00.570 Hello again! 00:01.080 --> 00:03.600 In this video, we are going to look at mathematical types. 00:05.190 --> 00:08.130 The first one we are going to look at, very briefly, is the valarray. 00:08.940 --> 00:10.740 This is similar to the vector. 00:11.220 --> 00:14.610 It is based on the arrays in the FORTRAN programming language. 00:15.270 --> 00:18.060 FORTRAN is one of the oldest computer programming languages. 00:18.120 --> 00:20.100 It goes back to the 1950s. 00:20.670 --> 00:24.720 It is specially designed for doing numerical work, and it is very good at it. 00:25.350 --> 00:26.850 So it is still used today. 00:28.140 --> 00:30.300 The valarray has a fixed size. 00:30.390 --> 00:36.450 The number of elements cannot change. And it has a much more limited interface than the standard vector. 00:36.990 --> 00:40.470 And it also has some extra syntax for doing numerical operations. 00:41.040 --> 00:43.890 So things like adding or dividing vectors, for example. 00:45.360 --> 00:47.220 However, it is not used very much. 00:47.760 --> 00:54.300 It is regarded as poorly designed, and compiler implementers are not really interested in optimizing it. 00:55.470 --> 01:01.050 If you want something like this, you would do better to use one of the third party libraries, and I have seen 01:01.050 --> 01:06.450 recommendations for Eigen, Blaze and Armadillo. And there are probably some other good ones as well. 01:09.100 --> 01:13.120 When we talked about overloading operators, we mentioned complex numbers. 01:13.810 --> 01:17.950 We wrote an addition operator for a very simple complex number class. 01:18.940 --> 01:26.350 C++ 11 now has a fully featured complex number class in the library, in the header. It is a 01:26.350 --> 01:31.750 template type, and the parameter can be any one of the built-in floating-point types. 01:32.290 --> 01:37.780 So we could use float, double or long double. Other types are "use at your own risk". 01:37.780 --> 01:40.390 They may not work properly or they may not even compile. 01:40.900 --> 01:42.220 So I do not recommend that. 01:44.380 --> 01:46.420 To create a complex number object, 01:46.420 --> 01:53.440 We give the type as the parameter so that is double, and then we part the real and imaginary parts as arguments 01:53.440 --> 01:54.310 to the constructor. 01:54.430 --> 02:01.210 So this creates a complex number whose real part is three and imaginary part is four. And there are member 02:01.210 --> 02:03.880 functions for getting the real and imaginary parts. 02:06.420 --> 02:09.180 We can do input and output with complex objects. 02:10.110 --> 02:13.320 The display format is the fairly obvious one. 02:13.590 --> 02:17.430 We have a pair of round brackets with the real and imaginary part inside them. 02:18.390 --> 02:22.530 We can also read in numbers from the keyboard, if they are entered in this format. 02:24.390 --> 02:25.450 So let's try that out. 02:25.470 --> 02:32.190 We are going to print out this complex number, and then we are going to try and read a complex number 02:32.190 --> 02:32.940 from the keyboard. 02:35.700 --> 02:43.140 So the complex number is printed out with real part three and imaginary part four. And let's enter one. 02:43.140 --> 02:43.620 So... 02:47.660 --> 02:48.020 See. 02:48.860 --> 02:56.090 So this has been interpreted as a complex number with real part 2.5, and imaginary part 1.9. 02:59.790 --> 03:05.400 We can do arithmetic and logical operations with complex numbers, and pretty much all of them are supported. 03:06.090 --> 03:12.150 So, for example, we can add together complex numbers, we can compare them for equality, and we can 03:12.510 --> 03:13.860 add integers introduced to them. 03:14.520 --> 03:19.440 If we do that, this will be interpreted as a complex number with real part one and imaginary part 03:19.460 --> 03:26.070 zero. So that will just affect the real part of the number. The increment and decrement operators 03:26.070 --> 03:27.150 are not supported. 03:27.930 --> 03:33.080 I suppose there is ambiguity. Does plus plus just mean adding one to the real 03:33.380 --> 03:37.770 part? Or does it mean adding one to the real part and the imaginary part? 03:38.400 --> 03:40.920 So I think they decided it would be easier not to allow 03:40.920 --> 03:41.160 that. 03:43.510 --> 03:45.160 So let's have some examples. 03:45.160 --> 03:48.430 So we are going to add together these two complex numbers. 03:49.600 --> 03:54.550 We are going to compare them for equality, and then we are going to add one to the first number. 03:58.420 --> 03:59.140 So there we are. 03:59.150 --> 04:02.170 One plus two gives four. Two plus 04:02.170 --> 04:05.230 four gives size. p and q are not equal. 04:05.950 --> 04:10.570 And if we add one to p, we get real part two, imaginary point two. 04:14.120 --> 04:19.160 If we try to call the increment operator, then we get a compiler error. 04:20.850 --> 04:23.370 "complex does not define this operator". 04:27.670 --> 04:36.550 In C++11, we have user-defined literals. In C++14, we get some literals in the library. We have this literal 04:36.960 --> 04:44.260 suffix "i", and this will return a complex object whose real part is zero and imaginary part is, whatever 04:44.260 --> 04:45.760 goes before this suffix. 04:46.990 --> 04:55.120 For example, if we have "2i", then this variable "s" will be a complex object with real part zero and imaginary 04:55.120 --> 04:59.860 part two. And we can also have "three plus 4i". 05:00.130 --> 05:04.510 So that will be a complex number with real part three and imaginary part four. 05:07.410 --> 05:11.850 So this means we can write complex numbers in the same way that we would write them in mathematics. 05:13.080 --> 05:18.140 So let's try this out, with this "2i" and this "three plus four i" variable. 05:19.680 --> 05:23.100 We also are going to do a compound addition with a literal. 05:26.150 --> 05:34.160 So there we are. "s" is real part zero, imaginary part two. "z" has real part three and imaginary part four. And 05:34.160 --> 05:41.840 adding "4i" to "p" gives real part one, imaginary part six. And then, finally, going a bit deeper into 05:41.840 --> 05:43.040 the maths side. 05:43.580 --> 05:46.520 If this is "above your pay grade", then do not worry about it. 05:47.870 --> 05:53.150 We can call abs() to get the magnitude, or the modulus, or the absolute value, whatever you want to call it. 05:53.780 --> 05:59.030 So that squares the real part and the imaginary part, adds them together, and takes the square root. 06:00.110 --> 06:06.470 We can call arg() to get the phase angle, in radians, and we can also get the complex conjugate. 06:06.950 --> 06:09.530 So that will invert the sign of the imaginary part. 06:12.270 --> 06:18.120 And we can also use exponential, power and trigonometric functions with complex objects. 06:21.170 --> 06:23.030 So let's quickly try that out. 06:27.550 --> 06:32.830 So the magnitude of "p" is five, the conjugate is three and minus four. 06:33.400 --> 06:35.410 And I will leave you to check the rest for yourself! 06:36.190 --> 06:36.490 Okay. 06:36.490 --> 06:37.570 So that is it for this video. 06:37.990 --> 06:38.800 I will see you next time. 06:38.800 --> 06:40.960 But until then, keep coding!