# complex

***

**1. Complex Number Arithmetic:**

```cpp
#include <complex>

int main() {
  std::complex<double> z1(1.0, 2.0), z2(3.0, 4.0);
  std::complex<double> sum = z1 + z2, prod = z1 * z2;
  std::cout << "Sum: " << sum << "\nProduct: " << prod << "\n";
  return 0;
}
```

**2. Roots of a Complex Quadratic Equation:**

```cpp
#include <complex>

int main() {
  double a = 1.0, b = -5.0, c = 6.0;
  std::complex<double> discriminant = b * b - 4 * a * c;
  std::complex<double> root1 = (-b + std::sqrt(discriminant)) / (2 * a);
  std::complex<double> root2 = (-b - std::sqrt(discriminant)) / (2 * a);
  std::cout << "Roots: " << root1 << ", " << root2 << "\n";
  return 0;
}
```

**3. Complex Fourier Transform:**

```cpp
#include <complex>
#include <vector>

std::vector<std::complex<double>> fft(std::vector<std::complex<double>> x) {
  const int N = x.size();
  if (N <= 1) return x;

  std::vector<std::complex<double>> even = fft(std::vector<std::complex<double>>(x.begin(), x.begin() + N / 2));
  std::vector<std::complex<double>> odd = fft(std::vector<std::complex<double>>(x.begin() + N / 2, x.end()));

  std::vector<std::complex<double>> y(N);
  for (int k = 0; k < N / 2; ++k) {
    const std::complex<double> w = std::polar(1.0, -2 * M_PI * k / N);
    y[k] = even[k] + w * odd[k];
    y[k + N / 2] = even[k] - w * odd[k];
  }

  return y;
}
```

**4. Complex Matrix Multiplication:**

```cpp
#include <complex>
#include <vector>

std::vector<std::vector<std::complex<double>>> cmult(std::vector<std::vector<std::complex<double>>> &A, std::vector<std::vector<std::complex<double>>> &B) {
  int m = A.size(), n = A[0].size(), p = B[0].size();
  std::vector<std::vector<std::complex<double>>> C(m, std::vector<std::complex<double>>(p, 0));
  for (int i = 0; i < m; ++i)
    for (int j = 0; j < p; ++j)
      for (int k = 0; k < n; ++k)
        C[i][j] += A[i][k] * B[k][j];
  return C;
}
```

**5. Complex Eigenvalues and Eigenvectors:**

```cpp
#include <complex>
#include <Eigen/Eigenvalues>

int main() {
  std::complex<double> A(1.0, 2.0), B(3.0, 4.0);
  Eigen::Matrix<std::complex<double>, 2, 2> M;
  M << A, B, std::conj(B), std::conj(A);
  Eigen::EigenSolver<Eigen::Matrix<std::complex<double>, 2, 2>> es(M);
  Eigen::VectorXcd eigenvalues = es.eigenvalues();
  Eigen::MatrixXcd eigenvectors = es.eigenvectors();
  std::cout << "Eigenvalues: " << eigenvalues << "\nEigenvectors:\n" << eigenvectors << "\n";
  return 0;
}
```

**6. Newton's Method for Complex Roots:**

```cpp
#include <complex>

std::complex<double> newton_root(std::complex<double> z0, std::complex<double> f(std::complex<double>)) {
  std::complex<double> z = z0;
  const int max_iter = 100;
  for (int i = 0; i < max_iter; ++i) {
    std::complex<double> df = (f(z + 1e-6) - f(z)) / 1e-6;
    z -= f(z) / df;
    if (std::abs(f(z)) < 1e-6) break;
  }
  return z;
}
```

**7. Solving Linear Systems with Complex Coefficients:**

```cpp
#include <complex>
#include <Eigen/LU>

int main() {
  std::complex<double> A[2][2] = {{1.0, 2.0}, {3.0, 4.0}};
  std::complex<double> b[2] = {5.0, 6.0};
  Eigen::MatrixXcd M(2, 2);
  M << A[0][0], A[0][1], A[1][0], A[1][1];
  Eigen::VectorXcd X = M.lu().solve(Eigen::VectorXcd(b));
  std::cout << "Solution: " << X << "\n";
  return 0;
}
```

**8. Complex Integration using Trapezoidal Rule:**

```cpp
#include <complex>

std::complex<double> trapezoidal_integration(
    std::complex<double> f(std::complex<double>),
    std::complex<double> a,
    std::complex<double> b,
    int n) {
  std::complex<double> h = (b - a) / n;
  std::complex<double> sum = 0;
  for (int i = 1; i < n; ++i) {
    sum += f(a + i * h);
  }
  return h * (0.5 * (f(a) + f(b)) + sum);
}
```

**9. Complex differentiation using numerical approximation:**

```cpp
#include <complex>

std::complex<double> numerical_derivative(
    std::complex<double> f(std::complex<double>),
    std::complex<double> x,
    double h) {
  return (f(x + h) - f(x - h)) / (2 * h);
}
```

**10. Complex Interpolation using Lagrange Interpolation:**

```cpp
#include <complex>

std::complex<double> lagrange_interpolation(
    std::vector<std::complex<double>> x,
    std::vector<std::complex<double>> y,
    std::complex<double> z) {
  std::complex<double> result = 0;
  for (int i = 0; i < x.size(); ++i) {
    std::complex<double> L = 1;
    for (int j = 0; j < x.size(); ++j) {
      if (i != j) {
        L *= (z - x[j]) / (x[i] - x[j]);
      }
    }
    result += L * y[i];
  }
  return result;
}
```

**11. Evaluation of Complex Polynomials using Horner's Rule:**

```cpp
#include <complex>

std::complex<double> horner_evaluation(
    std::vector<std::complex<double>> coefficients,
    std::complex<double> x) {
  std::complex<double> result = 0;
  for (int i = coefficients.size() - 1; i >= 0; --i) {
    result = result * x + coefficients[i];
  }
  return result;
}
```

**12. Complex Curve Fitting Using Least Squares:**

```cpp
#include <complex>
#include <Eigen/SVD>

std::complex<double> least_squares(
    std::vector<std::complex<double>> x,
    std::vector<std::complex<double>> y) {
  int m = x.size(), n = y.size();
  Eigen::MatrixXcd A(m, n);
  for (int i = 0; i < m; ++i) {
    for (int j = 0; j < n; ++j) {
      A(i, j) = std::pow(x[i], j);
    }
  }
  Eigen::VectorXcd b(m);
  for (int i = 0; i < m; ++i) {
    b[i] = y[i];
  }
  Eigen::JacobiSVD<Eigen::MatrixXcd> svd(A, Eigen::ComputeThinU | Eigen::ComputeThinV);
  Eigen::VectorXcd C = svd.solve(b);
  return C[0];
}
```

**13. Fast Fourier Transform for Complex Data:**

```cpp
#include <complex>
#include <vector>

std::vector<std::complex<double>> fft(std::vector<std::complex<double>> &x) {
  int n = x.size();
  if (n == 1) return x;

  std::vector<std::complex<double>> even(n / 2), odd(n / 2);
  for (int i = 0; i < n / 2; ++i) {
    even[i] = x[2 * i];
    odd[i] = x[2 * i + 1];
  }

  std::vector<std::complex<double>> y_even = fft(even);
  std::vector<std::complex<double>> y_odd = fft(odd);

  std::vector<std::complex<double>> y(n);
  for (int i = 0; i < n / 2; ++i) {
    std::complex<double> w = std::polar(1.0, -2 * M_PI * i / n);
    y[i] = y_even[i] + w * y_odd[i];
    y[i + n / 2] = y_even[i] - w * y_odd[i];
  }

  return y;
}
```

**14. Solving Partial Differential Equations using Finite Differences:**

```cpp
#include <complex>
#include <vector>

std::vector<std::vector<std::complex<double>>> solve_pde(
    std::complex<double> f(std::complex<double>, std::complex<double>),
    double x0,
    double y0,
    double h,
    int n) {
  std::vector<std::vector<std::complex<double>>> u(n + 1, std::vector<std::complex<double>>(n + 1));
  for (int i = 0; i <= n; ++i) {
    u[i][0] = f(x0 + i * h, y0);
    u[i][n] = f(x0 + i * h, y0 + h * n);
  }
  for (int j = 0; j <= n; ++j) {
    u[0][j] = f(x0, y0 + j * h);
    u[n][j] = f(x0 + h * n, y0 + j * h);
  }
  for (int i = 1; i < n; ++i) {
    for (int j = 1; j < n; ++j) {
      u[i][j] = (u[i - 1][j] + u[i + 1][j] + u[i][j - 1] + u[i][j + 1]) / 4.0;
    }
  }
  return u;
}
```

**15. Monte Carlo Simulation with Complex Numbers:**

```cpp
#include <complex>
#include <random>

std::complex<double> monte_carlo(
    std::complex<double> f(std::complex<double>),
    int n) {
  std::random_device rd;
  std::mt19937 gen(rd());
  std::uniform_real_distribution<double> dist(0.0, 1.0);

  std::complex<double> sum = 0;
  for (int i = 0; i < n; ++i) {
    std::complex<double> z(dist(gen), dist(gen));
    sum += f(z);
  }
  return sum / n;
}
```

**16. Complex Optimization using Gradient Descent:**

```cpp
#include <complex>

std::complex<double> gradient_descent(
    std::complex<double> f(std::complex<double>),
    std::complex<double> x0,
    double alpha,
    int max_iter) {
  std::complex<double> x = x0;
  for (int i = 0; i < max_iter; ++i) {
    std::complex<double> df = (f(x + 1e-6) - f(x)) / 1e-6;
    x -= alpha * df;
  }
  return x;
}
```

**17. Solving Systems of Nonlinear Equations with Complex Coefficients:**

```cpp
#include <complex>
#include <Eigen/IterativeLinearSolvers>

std::vector<std::complex<double>> solve_nonlinear(
    std::vector<std::complex<double>> f(std::vector<std::complex<double>>),
    std::vector<std::complex<double>> x0,
    double tol) {
  Eigen::VectorXcd x(x0);
  Eigen::BiCGSTAB<Eigen::VectorXcd> solver;
  solver.setTolerance(tol);
  Eigen::VectorXcd b = Eigen::VectorXcd::Zero(x0.size());
  x = solver.compute(f, b);
  return std::vector<std::complex<double>>(x.data(), x.data() + x.size());
}
```

**18. Generating Random Complex Numbers:**

```cpp
#include <complex>
#include <random>

std::complex<double> random_complex(double mean, double stddev) {
  std::random_device rd;
  std::mt19937 gen(rd());
  std::normal_distribution<double> dist(mean, stddev);

  double re = dist(gen);
  double im = dist(gen);
  return std::complex<double>(re, im);
}
```

**19. Complex Data Visualization using Matplotlib:**

```python
import matplotlib.pyplot as plt
import numpy as np

# Generate some complex data
z = np.random.randn(100) + 1j * np.random.randn(100)

# Plot the data
plt.scatter(z.real, z.imag)
plt.show()
```

**20. Complex Data Analysis using Pandas:**

```python
import pandas as pd

# Create a DataFrame with complex data
df = pd.DataFrame({
  "real": np.random.randn(100),
  "imag": np.random.randn(100)
})

# Perform some operations on the data
df["magnitude"] = np.abs(df["real"] + 1j * df["imag"])
df["phase"] = np.arctan2(df["imag"], df["real"])

# Print the results
print(df.head())
```

**21. Complex Number Operations in NumPy:**

```python
import numpy as np

# Create complex numbers
a = np.complex128(1, 2)
b = np.complex128(3, 4)

# Perform operations
c = a + b
d = a * b
e = np.conjugate(a)

# Print the results
print(c, d, e)
```

**22. Complex Fourier Transform in SciPy:**

```python
import scipy.fftpack

# Create a complex signal
x = np.random.randn(100) + 1j * np.random.randn(100)

# Compute the FFT
X = scipy.fftpack.fft(x)

# Plot the magnitude and phase of the FFT
plt.subplot(2, 1, 1)
plt.plot(np.abs(X))
plt.subplot(2, 1, 2)
plt.plot(np.angle(X))
plt.show()
```

**23. Complex Distance Calculation:**

```python
def complex_distance(z1, z2):
  """
  Calculates the distance between two complex numbers.

  Args:
    z1: The first complex number.
    z2: The second complex number.

  Returns:
    The distance between the two complex numbers.
  """

  return np.sqrt((z1.real - z2.real)**2 + (z1.imag - z2.imag)**2)
```

**24. Complex Number Sorting:**

```python
def complex_sort(z):
  """
  Sorts a list of complex numbers by their magnitude.

  Args:
    z: The list of complex numbers to sort.

  Returns:
    The sorted list of complex numbers.
  """

  return sorted(z, key=lambda x: abs(x))
```

**25. Complex Number Interpolation:**

```python
def complex_interpolate(z1, z2, t):
  """
  Interpolates between two complex numbers.

  Args:
    z1: The first complex number.
    z2: The second complex number.
    t: The interpolation parameter (between 0 and 1).

  Returns:
    The interpolated complex number.
  """

  return z1 + t * (z2 - z1)
```

**26. Complex Number Differentiation:**

```python
def complex_derivative(z):
  """
  Computes the derivative of a complex number.

  Args:
    z: The complex number to differentiate.

  Returns:
    The derivative of the complex number.
  """

  return 1
```

**27. Complex Number Integration:**

```python
def complex_integral(z, a, b):
  """
  Computes the integral of a complex number over an interval.

  Args:
    z: The complex number to integrate.
    a: The lower bound of the interval.
    b: The upper bound of the interval.

  Returns:
    The integral of the complex number over the interval.
  """

  return (b - a) * z
```

**28. Complex Number Series:**

```python
def complex_series(n):
  """
  Computes the sum of the first n terms of the complex series.

  Args:
    n: The number of terms to sum.

  Returns:
    The sum of the first n terms of the complex series.
  """

  return sum(1 / (2 * i + 1) * 1j for i in range(n))
```

**29. Complex Number Trigonometry:**

```python
def complex_sin(z):
  """
  Computes the sine of a complex number.

  Args:
    z: The complex number to compute the sine of.

  Returns:
    The sine of the complex number.
  """

  return (np.exp(1j * z) - np.exp(-1j * z)) / (2j)
```

**30. Complex Number Exponentiation:**

```python
def complex_exp(z):
  """
  Computes the exponential of a complex number.

  Args:
    z: The complex number to compute the exponential of.

  Returns:
    The exponential of the complex number.
  """

  return np.exp(z)
```

**31. Complex Number Logarithm:**

```python
def complex_log(z):
  """
  Computes the logarithm of a complex number.

  Args:
    z: The complex number to compute the logarithm of.

  Returns:
    The logarithm of the complex number.
  """

  return np.log(z)
```

**32. Complex Number Power:**

```python
def complex_pow(z, n):
  """
  Computes the power of a complex number.

  Args:
    z: The complex number to compute the power of.
    n: The power to raise the complex number to.

  Returns:
    The power of the complex number.
  """

  return z ** n
```

**33. Complex Number Conjugate:**

```python
def complex_conjugate(z):
  """
  Computes the conjugate of a complex number.

  Args:
    z: The complex number to compute the conjugate of.

  Returns:
    The conjugate of the complex number.
  """

  return np.conj(z)
```

**34. Complex Number Reciprocal:**

```python
def complex_reciprocal(z):
  """
  Computes the reciprocal of a complex number.

  Args:
    z: The complex number to compute the reciprocal of.

  Returns:
    The reciprocal of the complex number.
  """

  return 1 / z
```

**35. Complex Number Root:**

```python
def complex_root(z, n):
  """
  Computes the nth root of a complex number.

  Args:
    z: The complex number to compute the root of.
    n: The order of the root.

  Returns:
    The nth root of the complex number.
  """

  return z ** (1 / n)
```

**36. Complex Number Division:**

```python
def complex_divide(z1, z2):
  """
  Divides one complex number by another.

  Args:
    z1: The dividend complex number.
    z2: The divisor complex number.

  Returns:
    The quotient complex number.
  """

  return z1 / z2
```

**37. Complex Number Multiplication:**

```python
def complex_multiply(z1, z2):
  """
  Multiplies two complex numbers.

  Args:
    z1: The first complex number.
    z2: The second complex number.

  Returns:
    The product complex number.
  """

  return z1 * z2
```

**38. Complex Number Addition:**

```python
def complex_add(z1, z2):
  """
  Adds two complex numbers.

  Args:
    z1: The first complex number.
    z2: The second complex number.

  Returns:
    The sum complex number.
  """

  return z1 + z2
```

**39. Complex Number Subtraction:**

```python
def complex_subtract(z1, z2):
  """
  Subtracts one complex number from another.

  Args:
    z1: The minuend complex number.
    z2: The subtrahend complex number.

  Returns:
    The difference complex number.
  """

  return z1 - z2
```

**40. Complex Number Absolute Value:**

```python
def complex_abs(z):
  """
  Computes the absolute value of a complex number.

  Args:
    z: The complex number to compute the absolute value of.

  Returns:
    The absolute value of the complex number.
  """

  return abs(z)
```

**41. Complex Number Phase:**

```python
def complex_phase(z):
  """
  Computes the phase of a complex number.

  Args:
    z: The complex number to compute the phase of.

  Returns:
    The phase of the complex number.
  """

  return np.angle(z)
```

**42. Complex Number Real Part:**

```python
def complex_real(z):
  """
  Gets the real part of a complex number.

  Args:
    z: The complex number to get the real part of.

  Returns:
    The real part of the complex number.
  """

  return z.real
```

**43. Complex Number Imaginary Part:**

```python
def complex_imag(z):
  """
  Gets the imaginary part of a complex number.

  Args:
    z: The complex number to get the imaginary part of.

  Returns:
    The imaginary part of the complex number.
  """

  return z.imag
```

**44. Complex Number Comparison:**

```python
def complex_compare(z1, z2):
  """
  Compares two complex numbers.

  Args:
    z1: The first complex number.
    z2: The second complex number.

  Returns:
    True if the two complex numbers are equal, False otherwise.
  """

  return z1 == z2
```

**45. Complex Number Conversion from String:**

```python
def complex_from_string(s):
  """
  Converts a string to a complex number.

  Args:
    s: The string to convert to a complex number.

  Returns:
    The complex number represented by the string.
  """

  return complex(s)
```

**46. Complex Number Conversion to String:**

```python
def complex_to_string(z):
  """
  Converts a complex number to a string.

  Args:
    z: The complex number to convert to a string.

  Returns:
    The string representation of the complex number.
  """

  return str(z)
```

**47. Complex Number Rounding:**

```python
def complex_round(z, n):
  """
  Rounds a complex number to the specified number of decimal places.

  Args:
    z: The complex number to round.
    n: The number of decimal places to round to.

  Returns:
    The rounded complex number.
  """

  return round(z, n)
```

**48. Complex Number Format String:**

```python
def complex_format(z, format_string):
  """
  Formats a complex number using a format string.

  Args:
    z: The complex number to format.
    format_string: The format string to use.

  Returns:
    The formatted complex number.
  """

  return format(z, format_string)
```

**49. Complex Number Hash Function:**

```python
def complex_hash(z):
  """
  Computes the hash value of a complex number.

  Args:
    z: The complex number to compute the hash value of.

  Returns:
    The hash value of the complex number.
  """

  return hash(z)
```

**50. Complex Number Copy:**

```python
def complex_copy(z):
  """
  Copies a complex number.

  Args:
    z: The complex number to copy.

  Returns:
    A copy of the complex number.
  """

  return complex(z)
```