# Standard library: Numerics¶

The standard library provides support for common numeric operations on floating-point types as well as on complex types and matrices. In the sections below, we present a brief introduction to these numeric operations.

## Elementary Functions¶

The Ada.Numerics.Elementary_Functions package provides common operations for floating-point types, such as square root, logarithm, and the trigonometric functions (e.g., sin, cos). For example:

with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics; use Ada.Numerics; with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions; procedure Show_Elem_Math is X : Float; begin X := 2.0; Put_Line ("Square root of " & Float'Image (X) & " is " & Float'Image (Sqrt (X))); X := e; Put_Line ("Natural log of " & Float'Image (X) & " is " & Float'Image (Log (X))); X := 10.0 ** 6.0; Put_Line ("Log_10 of " & Float'Image (X) & " is " & Float'Image (Log (X, 10.0))); X := 2.0 ** 8.0; Put_Line ("Log_2 of " & Float'Image (X) & " is " & Float'Image (Log (X, 2.0))); X := Pi; Put_Line ("Cos of " & Float'Image (X) & " is " & Float'Image (Cos (X))); X := -1.0; Put_Line ("Arccos of " & Float'Image (X) & " is " & Float'Image (Arccos (X))); end Show_Elem_Math;

Here we use the standard e and Pi constants from the Ada.Numerics package.

The Ada.Numerics.Elementary_Functions package provides operations for the Float type. Similar packages are available for Long_Float and Long_Long_Float types. For example, the Ada.Numerics.Long_Elementary_Functions package offers the same set of operations for the Long_Float type. In addition, the Ada.Numerics.Generic_Elementary_Functions package is a generic version of the package that you can instantiate for custom floating-point types. In fact, the Elementary_Functions package can be defined as follows:

package Elementary_Functions is new


## Random Number Generation¶

The Ada.Numerics.Float_Random package provides a simple random number generator for the range between 0.0 and 1.0. To use it, declare a generator G, which you pass to Random. For example:

with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Float_Random; use Ada.Numerics.Float_Random; procedure Show_Float_Random_Num is G : Generator; X : Uniformly_Distributed; begin Reset (G); Put_Line ("Some random numbers between " & Float'Image (Uniformly_Distributed'First) & " and " & Float'Image (Uniformly_Distributed'Last) & ":"); for I in 1 .. 15 loop X := Random (G); Put_Line (Float'Image (X)); end loop; end Show_Float_Random_Num;

The standard library also includes a random number generator for discrete numbers, which is part of the Ada.Numerics.Discrete_Random package. Since it's a generic package, you have to instantiate it for the desired discrete type. This allows you to specify a range for the generator. In the following example, we create an application that displays random integers between 1 and 10:

with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Discrete_Random; procedure Show_Discrete_Random_Num is subtype Random_Range is Integer range 1 .. 10; package R is new Ada.Numerics.Discrete_Random (Random_Range); use R; G : Generator; X : Random_Range; begin Reset (G); Put_Line ("Some random numbers between " & Integer'Image (Random_Range'First) & " and " & Integer'Image (Random_Range'Last) & ":"); for I in 1 .. 15 loop X := Random (G); Put_Line (Integer'Image (X)); end loop; end Show_Discrete_Random_Num;

Here, package R is instantiated with the Random_Range type, which has a constrained range between 1 and 10. This allows us to control the range used for the random numbers. We could easily modify the application to display random integers between 0 and 20 by changing the specification of the Random_Range type. We can also use floating-point or fixed-point types.

## Complex Types¶

The Ada.Numerics.Complex_Types package provides support for complex number types and the Ada.Numerics.Complex_Elementary_Functions package provides support for common operations on complex number types, similar to the Ada.Numerics.Elementary_Functions package. Finally, you can use the Ada.Text_IO.Complex_IO package to perform I/O operations on complex numbers. In the following example, we declare variables of the Complex type and initialize them using an aggregate:

with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics; use Ada.Numerics; with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types; with Ada.Numerics.Complex_Elementary_Functions; use Ada.Numerics.Complex_Elementary_Functions; with Ada.Text_IO.Complex_IO; procedure Show_Elem_Math is package C_IO is new Ada.Text_IO.Complex_IO (Complex_Types); use C_IO; X, Y : Complex; R, Th : Float; begin X := (2.0, -1.0); Y := (3.0, 4.0); Put (X); Put (" * "); Put (Y); Put (" is "); Put (X * Y); New_Line; New_Line; R := 3.0; Th := Pi / 2.0; X := Compose_From_Polar (R, Th); -- Alternatively: -- X := R * Exp ((0.0, Th)); -- X := R * e ** Complex'(0.0, Th); Put ("Polar form: " & Float'Image (R) & " * e**(i * " & Float'Image (Th) & ")"); New_Line; Put ("Modulus of "); Put (X); Put (" is "); Put (Float'Image (abs (X))); New_Line; Put ("Argument of "); Put (X); Put (" is "); Put (Float'Image (Argument (X))); New_Line; New_Line; Put ("Sqrt of "); Put (X); Put (" is "); Put (Sqrt (X)); New_Line; end Show_Elem_Math;

As we can see from this example, all the common operators, such as * and +, are available for complex types. You also have typical operations on complex numbers, such as Argument and Exp. In addition to initializing complex numbers in the cartesian form using aggregates, you can do so from the polar form by calling the Compose_From_Polar function.

The Ada.Numerics.Complex_Types and Ada.Numerics.Complex_Elementary_Functions packages provide operations for the Float type. Similar packages are available for Long_Float and Long_Long_Float types. In addition, the Ada.Numerics.Generic_Complex_Types and Ada.Numerics.Generic_Complex_Elementary_Functions packages are generic versions that you can instantiate for custom or pre-defined floating-point types. For example:

with Ada.Numerics.Generic_Complex_Types;

procedure Show_Elem_Math is

package Complex_Types is new
use Complex_Types;

package Elementary_Functions is new
use Elementary_Functions;

package C_IO is new Ada.Text_IO.Complex_IO (Complex_Types);
use C_IO;

X, Y  : Complex;
R, Th : Float;


## Vector and Matrix Manipulation¶

The Ada.Numerics.Real_Arrays package provides support for vectors and matrices. It includes common matrix operations such as inverse, determinant, eigenvalues in addition to simpler operators such as matrix addition and multiplication. You can declare vectors and matrices using the Real_Vector and Real_Matrix types, respectively.

The following example uses some of the operations from the Ada.Numerics.Real_Arrays package:

with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays; procedure Show_Matrix is procedure Put_Vector (V : Real_Vector) is begin Put (" ("); for I in V'Range loop Put (Float'Image (V (I)) & " "); end loop; Put_Line (")"); end Put_Vector; procedure Put_Matrix (M : Real_Matrix) is begin for I in M'Range (1) loop Put (" ("); for J in M'Range (2) loop Put (Float'Image (M (I, J)) & " "); end loop; Put_Line (")"); end loop; end Put_Matrix; V1 : Real_Vector := (1.0, 3.0); V2 : Real_Vector := (75.0, 11.0); M1 : Real_Matrix := ((1.0, 5.0, 1.0), (2.0, 2.0, 1.0)); M2 : Real_Matrix := ((31.0, 11.0, 10.0), (34.0, 16.0, 11.0), (32.0, 12.0, 10.0), (31.0, 13.0, 10.0)); M3 : Real_Matrix := ((1.0, 2.0), (2.0, 3.0)); begin Put_Line ("V1"); Put_Vector (V1); Put_Line ("V2"); Put_Vector (V2); Put_Line ("V1 * V2 ="); Put_Line (" " & Float'Image (V1 * V2)); Put_Line ("V1 * V2 ="); Put_Matrix (V1 * V2); New_Line; Put_Line ("M1"); Put_Matrix (M1); Put_Line ("M2"); Put_Matrix (M2); Put_Line ("M2 * Transpose(M1) ="); Put_Matrix (M2 * Transpose (M1)); New_Line; Put_Line ("M3"); Put_Matrix (M3); Put_Line ("Inverse (M3) ="); Put_Matrix (Inverse (M3)); Put_Line ("abs Inverse (M3) ="); Put_Matrix (abs Inverse (M3)); Put_Line ("Determinant (M3) ="); Put_Line (" " & Float'Image (Determinant (M3))); Put_Line ("Solve (M3, V1) ="); Put_Vector (Solve (M3, V1)); Put_Line ("Eigenvalues (M3) ="); Put_Vector (Eigenvalues (M3)); New_Line; end Show_Matrix;

Matrix dimensions are automatically determined from the aggregate used for initialization when you don't specify them. You can, however, also use explicit ranges. For example:

M1       : Real_Matrix (1 .. 2, 1 .. 3) :=
((1.0, 5.0, 1.0),
(2.0, 2.0, 1.0));


The Ada.Numerics.Real_Arrays package implements operations for the Float type. Similar packages are available for Long_Float and Long_Long_Float types. In addition, the Ada.Numerics.Generic_Real_Arrays package is a generic version that you can instantiate with custom floating-point types. For example, the Real_Arrays package can be defined as follows:

package Real_Arrays is new