math Namespace Reference

for M_PI definition and pow() More...

Functions
template<typename T >
T	square_area (T length)
	area of a square (l * l) More...
template<typename T >
T	rect_area (T length, T width)
	area of a rectangle (l * w) More...
template<typename T >
T	triangle_area (T base, T height)
	area of a triangle (b * h / 2) More...
template<typename T >
T	circle_area (T radius)
	area of a circle (pi More...
template<typename T >
T	parallelogram_area (T base, T height)
	area of a parallelogram (b * h) More...
template<typename T >
T	cube_surface_area (T length)
	surface area of a cube ( 6 * (l More...
template<typename T >
T	sphere_surface_area (T radius)
	surface area of a sphere ( 4 * pi * r^2) More...
template<typename T >
T	cylinder_surface_area (T radius, T height)
	surface area of a cylinder (2 * pi * r * h + 2 * pi * r^2) More...
double	integral_approx (double lb, double ub, const std::function< double(double)> &func, double delta=.0001)
	Computes integral approximation. More...
void	test_eval (double approx, double expected, double threshold)
	Wrapper to evaluate if the approximated value is within .XX% threshold of the exact value. More...
uint64_t	largestPower (uint32_t n, const uint16_t &p)
	Function to calculate largest power. More...
uint64_t	lcmSum (const uint16_t &num)
bool	magic_number (const uint64_t &n)
uint64_t	power (uint64_t a, uint64_t b, uint64_t c)
	This function calculates a raised to exponent b under modulo c using modular exponentiation. More...
template<class T >
T	n_choose_r (T n, T r)
	This is the function implementation of \( \binom{n}{r} \). More...
void	power_of_two (int n)
	Function to test above algorithm. More...
uint64_t	binomialCoeffSum (uint64_t n)
template<typename T >
T	cube_volume (T length)
	The volume of a cube More...
template<typename T >
T	rect_prism_volume (T length, T width, T height)
	The volume of a rectangular prism. More...
template<typename T >
T	cone_volume (T radius, T height, double PI=3.14)
	The volume of a cone More...
template<typename T >
T	triangle_prism_volume (T base, T height, T depth)
	The volume of a triangular prism. More...
template<typename T >
T	pyramid_volume (T length, T width, T height)
	The volume of a pyramid More...
template<typename T >
T	sphere_volume (T radius, double PI=3.14)
	The volume of a sphere More...
template<typename T >
T	cylinder_volume (T radius, T height, double PI=3.14)
	The volume of a cylinder More...

Detailed Description

for M_PI definition and pow()

for std::cin and std::cout

for std::cout

for io operations

Evaluate recurrence relation using matrix exponentiation.

for assert

Math algorithms.

for std::vector

for IO operations

for assert for uint16_t datatype for IO operations

Mathematical algorithms

for assert for int32_t type for atoi

Mathematical algorithms

for assert for std::cin and std::cout

Mathematical algorithms

for assert for mathematical functions for passing in functions

Mathematical functions

for math functions for fixed size data types for time to initialize rng for function pointers for std::cout for random number generation for std::vector

for std::cin and std::cout

Mathematical algorithms

Given a recurrence relation; evaluate the value of nth term. For e.g., For fibonacci series, recurrence series is f(n) = f(n-1) + f(n-2) where f(0) = 0 and f(1) = 1. Note that the method used only demonstrates recurrence relation with one variable (n), unlike nCr problem, since it has two (n, r)

Algorithm

This problem can be solved using matrix exponentiation method.

See also

here for simple number exponentiation algorithm or explaination here.

Author

Ashish Daulatabad for assert for IO operations for std::vector STL

Mathematical algorithms

for assert

Mathematical algorithms

for std::is_equal, std::swap for assert for IO operations

Mathematical algorithms

for assert for io operations

Mathematical algorithms

Mathematical algorithms

for assert for std::pow for std::uint32_t

Mathematical algorithms

Function Documentation

binomialCoeffSum()

uint64_t math::binomialCoeffSum

(

uint64_t

n

)

Function to calculate sum of binomial coefficients

Parameters

n

number

Returns

Sum of binomial coefficients of number

                                      {
    // Calculating 2^n
    return (1 << n);
}

circle_area()

template<typename T >

T math::circle_area

(

T

radius

)

area of a circle (pi

• r^2)

  Parameters

  radius

  is the radius of the circle

  Returns

  area of the circle

                        {
    return M_PI * pow(radius, 2);
}

cone_volume()

template<typename T >

T math::cone_volume	(	T	radius,
		T	height,
		double	PI = 3.14
	)

The volume of a cone

Parameters

radius	The radius of the base circle
height	The height of the cone
PI	The definition of the constant PI

Returns

The volume of the cone

                                                    {
    return std::pow(radius, 2) * PI * height / 3;
}

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cube_surface_area()

template<typename T >

T math::cube_surface_area

(

T

length

)

surface area of a cube ( 6 * (l

• l))

  Parameters

  length

  is the length of the cube

  Returns

  surface area of the cube

                              {
    return 6 * length * length;
}

cube_volume()

template<typename T >

T math::cube_volume

(

T

length

)

The volume of a cube

Parameters

length

The length of the cube

Returns

The volume of the cube

                        {
    return std::pow(length, 3);
}

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cylinder_surface_area()

template<typename T >

T math::cylinder_surface_area	(	T	radius,
		T	height
	)

surface area of a cylinder (2 * pi * r * h + 2 * pi * r^2)

Parameters

radius	is the radius of the cylinder
height	is the height of the cylinder

Returns

surface area of the cylinder

                                            {
    return 2 * M_PI * radius * height + 2 * M_PI * pow(radius, 2);
}

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cylinder_volume()

template<typename T >

T math::cylinder_volume	(	T	radius,
		T	height,
		double	PI = 3.14
	)

The volume of a cylinder

Parameters

radius	The radius of the base circle
height	The height of the cylinder
PI	The definition of the constant PI

Returns

The volume of the cylinder

                                                        {
    return PI * std::pow(radius, 2) * height;
}

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integral_approx()

double math::integral_approx	(	double	lb,
		double	ub,
		const std::function< double(double)> &	func,
		double	delta = .0001
	)

Computes integral approximation.

Parameters

lb	lower bound
ub	upper bound
func	function passed in
delta

Returns

integral approximation of function from [lb, ub]

                                             {
    double result = 0;
    uint64_t numDeltas = static_cast<uint64_t>((ub - lb) / delta);
    for (int i = 0; i < numDeltas; i++) {
        double begin = lb + i * delta;
        double end = lb + (i + 1) * delta;
        result += delta * (func(begin) + func(end)) / 2;
    }
    return result;
}

largestPower()

uint64_t math::largestPower	(	uint32_t	n,
		const uint16_t &	p
	)

Function to calculate largest power.

Parameters

n	number
p	prime number

Returns

largest power

    {  
        // Initialize result  
        int x = 0;  
  
        // Calculate result 
        while (n)  
        {  
            n /= p;  
            x += n;  
        }  
        return x;  
    }

lcmSum()

uint64_t math::lcmSum

(

const uint16_t &

num

)

Function to compute sum of euler totients in sumOfEulerTotient vector

Parameters

num	input number

Returns

int Sum of LCMs, i.e. ∑LCM(i, num) from i = 1 to num

                                     {
    uint64_t i = 0, j = 0;
    std::vector<uint64_t> eulerTotient(num + 1);
    std::vector<uint64_t> sumOfEulerTotient(num + 1);
 
    // storing initial values in eulerTotient vector
    for (i = 1; i <= num; i++) {
        eulerTotient[i] = i;
    }
 
    // applying totient sieve
    for (i = 2; i <= num; i++) {
        if (eulerTotient[i] == i) {
            for (j = i; j <= num; j += i) {
                eulerTotient[j] = eulerTotient[j] / i;
                eulerTotient[j] = eulerTotient[j] * (i - 1);
            }
        }
    }
 
    // computing sum of euler totients
    for (i = 1; i <= num; i++) {
        for (j = i; j <= num; j += i) {
            sumOfEulerTotient[j] += eulerTotient[i] * i;
        }
    }
 
    return ((sumOfEulerTotient[num] + 1) * num) / 2;
}

magic_number()

bool math::magic_number

(

const uint64_t &

n

)

Function to check if the given number is magic number or not.

Parameters

n	number to be checked.

Returns

if number is a magic number, returns true, else false.

                                     {
    if (n <= 0) {
        return false;
    }
    // result stores the modulus of @param n with 9
    uint64_t result = n % 9;
    // if result is 1 then the number is a magic number else not
    if (result == 1) {
        return true;
    } else {
        return false;
    }
}

n_choose_r()

template<class T >

T math::n_choose_r	(	T	n,
		T	r
	)

This is the function implementation of \( \binom{n}{r} \).

We are calculating the ans with iterations instead of calculating three different factorials. Also, we are using the fact that \( \frac{n!}{r! (n-r)!} = \frac{(n - r + 1) \times \cdots \times n}{1 \times \cdots \times r} \)

Template Parameters

T	Only for integer types such as long, int_64 etc

Parameters

n	\( n \) in \( \binom{n}{r} \)
r	\( r \) in \( \binom{n}{r} \)

Returns

ans \( \binom{n}{r} \)

                       {
    if (r > n / 2) {
        r = n - r;  // Because of the fact that  nCr(n, r) == nCr(n, n - r)
    }
    T ans = 1;
    for (int i = 1; i <= r; i++) {
        ans *= n - r + i;
        ans /= i;
    }
    return ans;
}

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parallelogram_area()

template<typename T >

T math::parallelogram_area	(	T	base,
		T	height
	)

area of a parallelogram (b * h)

Parameters

base	is the length of the bottom side of the parallelogram
height	is the length of the tallest point in the parallelogram

Returns

area of the parallelogram

                                       {
    return base * height;
}

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power()

uint64_t math::power	(	uint64_t	a,
		uint64_t	b,
		uint64_t	c
	)

This function calculates a raised to exponent b under modulo c using modular exponentiation.

Parameters

a	integer base
b	unsigned integer exponent
c	integer modulo

Returns

a raised to power b modulo c

Initialize the answer to be returned

Update a if it is more than or equal to c

In case a is divisible by c;

If b is odd, multiply a with answer

b must be even now

b = b/2

                                                   {
    uint64_t ans = 1;  /// Initialize the answer to be returned
    a = a % c;         /// Update a if it is more than or equal to c
    if (a == 0) {
        return 0;  /// In case a is divisible by c;
    }
    while (b > 0) {
        /// If b is odd, multiply a with answer
        if (b & 1) {
            ans = ((ans % c) * (a % c)) % c;
        }
        /// b must be even now
        b = b >> 1;  /// b = b/2
        a = ((a % c) * (a % c)) % c;
    }
    return ans;
}

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power_of_two()

void math::power_of_two

(

int

n

)

Function to test above algorithm.

Parameters

n	description

Returns

void

This function finds whether a number is power of 2 or not

Parameters

n	value for which we want to check prints the result, as "Yes, the number n is a power of 2" or "No, the number is not a power of 2" without quotes

result stores the bitwise and of n and n-1

                         {
    /**
     * This function finds whether a number is power of 2 or not
     * @param n value for which we want to check
     * prints the result, as "Yes, the number n is a power of 2" or
     * "No, the number is not a power of 2" without quotes
     */
    /// result stores the
    /// bitwise and of n and n-1
    int result = n & (n - 1);
    if (result == 0) {
        std::cout << "Yes, the number " << n << " is a power of 2";
    } else {
        std::cout << "No, the number " << n << " is not a power of 2";
    }
}

pyramid_volume()

template<typename T >

T math::pyramid_volume	(	T	length,
		T	width,
		T	height
	)

The volume of a pyramid

Parameters

length	The length of the base shape (or base for triangles)
width	The width of the base shape (or height for triangles)
height	The height of the pyramid

Returns

The volume of the pyramid

                                              {
    return length * width * height / 3;
}

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rect_area()

template<typename T >

T math::rect_area	(	T	length,
		T	width
	)

area of a rectangle (l * w)

Parameters

length	is the length of the rectangle
width	is the width of the rectangle

Returns

area of the rectangle

                               {
    return length * width;
}

rect_prism_volume()

template<typename T >

T math::rect_prism_volume	(	T	length,
		T	width,
		T	height
	)

The volume of a rectangular prism.

Parameters

length	The length of the base rectangle
width	The width of the base rectangle
height	The height of the rectangular prism

Returns

The volume of the rectangular prism

                                                 {
    return length * width * height;
}

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sphere_surface_area()

template<typename T >

T math::sphere_surface_area

(

T

radius

)

surface area of a sphere ( 4 * pi * r^2)

Parameters

radius

is the radius of the sphere

Returns

surface area of the sphere

                                {
    return 4 * M_PI * pow(radius, 2);
}

sphere_volume()

template<typename T >

T math::sphere_volume	(	T	radius,
		double	PI = 3.14
	)

The volume of a sphere

Parameters

radius	The radius of the sphere
PI	The definition of the constant PI

Returns

The volume of the sphere

                                            {
    return PI * std::pow(radius, 3) * 4 / 3;
}

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square_area()

template<typename T >

T math::square_area

(

T

length

)

area of a square (l * l)

Parameters

length

is the length of the square

Returns

area of square

                        {
    return length * length;
}

test_eval()

void math::test_eval	(	double	approx,
		double	expected,
		double	threshold
	)

Wrapper to evaluate if the approximated value is within .XX% threshold of the exact value.

Parameters

approx	aprroximate value
exact	expected value
threshold	values from [0, 1)

                                                                 {
    assert(approx >= expected * (1 - threshold));
    assert(approx <= expected * (1 + threshold));
}

triangle_area()

template<typename T >

T math::triangle_area	(	T	base,
		T	height
	)

area of a triangle (b * h / 2)

Parameters

base	is the length of the bottom side of the triangle
height	is the length of the tallest point in the triangle

Returns

area of the triangle

                                  {
    return base * height / 2;
}

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triangle_prism_volume()

template<typename T >

T math::triangle_prism_volume	(	T	base,
		T	height,
		T	depth
	)

The volume of a triangular prism.

Parameters

base	The length of the base triangle
height	The height of the base triangles
depth	The depth of the triangular prism (the height of the whole prism)

Returns

The volume of the triangular prism

                                                   {
    return base * height * depth / 2;
}

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