{"id":1771,"date":"2024-03-18T15:52:43","date_gmt":"2024-03-18T10:22:43","guid":{"rendered":"https:\/\/moodle.sit.ac.in\/blog\/?p=1771"},"modified":"2024-05-13T23:33:13","modified_gmt":"2024-05-13T18:03:13","slug":"mathematics-ii-for-cse-stream-lab-component","status":"publish","type":"post","link":"https:\/\/moodle.sit.ac.in\/blog\/mathematics-ii-for-cse-stream-lab-component\/","title":{"rendered":"Mathematics-II for CSE stream Lab Component"},"content":{"rendered":"\n

(Course Code: BMATS201)<\/mark> solutions using Python Programming<\/p>\n\n\n\n

<\/p>\n\n\n\n

Hi everyone! This blog provides solutions for the lab component of the newly introduced subject “Mathematics-II for Computer Science and Engineering stream<\/strong>” (BMATS201) for II semester students of VTU. I have provided solutions for the lab exercises using Python. All the programs shown below have been tested on Python 3.11.7<\/strong> on Ubuntu 22.04<\/strong> OS using Anaconda Distribution<\/strong>. <\/p>\n\n\n\n

Installation Instructions<\/h2>\n\n\n\n

A detailed procedure of installing ANACONDA <\/strong>distribution and necessary tools for this is provided in my other blog listed below. https:\/\/moodle.sit.ac.in\/blog\/setting-up-anaconda-python-programming-environment\/<\/a><\/p>\n\n\n\n

Questions<\/h2>\n\n\n\n
    \n
  1. Program to compute area, surface area, volume and centre of gravity<\/a><\/li>\n\n\n\n
  2. Evaluation of improper integrals<\/a><\/li>\n\n\n\n
  3. Finding gradient, divergent, curl and their geometrical interpretation<\/a><\/li>\n\n\n\n
  4. Basis and Dimension for a vector space and Graphical representation of linear transformation<\/a><\/li>\n\n\n\n
  5. Computing the inner product and orthogonality<\/a><\/li>\n\n\n\n
  6. Solution of algebraic and transcendental equations by Ramanujan\u2019s, Regula-Falsi and Newton-Raphson method<\/a><\/li>\n\n\n\n
  7. Interpolation\/Extrapolation using Newton\u2019s forward and backward difference formula<\/a><\/li>\n\n\n\n
  8. Computation of area under the curve using Trapezoidal, Simpson\u2019s (1\/3)rd and (3\/8)th rule<\/a><\/li>\n\n\n\n
  9. Solution of ODE of first order and first degree by Taylor\u2019s series and Modified Euler\u2019s method<\/a><\/li>\n\n\n\n
  10. Solution of ODE of first order and first degree by Runge-Kutta 4th order and Milne\u2019s predictor-corrector method<\/a><\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n

    1) Program to compute area, surface area, volume and centre of gravity<\/h2>\n\n\n\n

    Below is a Python program that computes the area, surface area, volume, and center of gravity of a given 3D shape. This example specifically targets a sphere, but you can adapt it for other shapes by defining appropriate functions for area, surface area, volume, and center of gravity calculations.<\/p>\n\n\n\n

    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> math<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>sphere_area<\/span>(<\/span>radius<\/span>):<\/span><\/span>\n    <\/span>return<\/span> <\/span>4<\/span> <\/span>*<\/span> math.pi <\/span>*<\/span> radius <\/span>**<\/span> <\/span>2<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>sphere_surface_area<\/span>(<\/span>radius<\/span>):<\/span><\/span>\n    <\/span>return<\/span> <\/span>4<\/span> <\/span>*<\/span> math.pi <\/span>*<\/span> radius <\/span>**<\/span> <\/span>2<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>sphere_volume<\/span>(<\/span>radius<\/span>):<\/span><\/span>\n    <\/span>return<\/span> (<\/span>4<\/span>\/<\/span>3<\/span>) <\/span>*<\/span> math.pi <\/span>*<\/span> radius <\/span>**<\/span> <\/span>3<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>sphere_center_of_gravity<\/span>(<\/span>radius<\/span>):<\/span><\/span>\n    <\/span>return<\/span> (<\/span>3<\/span>\/<\/span>8<\/span>) <\/span>*<\/span> radius<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>main<\/span>():<\/span><\/span>\n    radius <\/span>=<\/span> <\/span>float<\/span>(<\/span>input<\/span>(<\/span>"<\/span>Enter the radius of the sphere: <\/span>"<\/span>))<\/span><\/span>\n<\/span>\n    area <\/span>=<\/span> sphere_area(radius)<\/span><\/span>\n    surface_area <\/span>=<\/span> sphere_surface_area(radius)<\/span><\/span>\n    volume <\/span>=<\/span> sphere_volume(radius)<\/span><\/span>\n    center_of_gravity <\/span>=<\/span> sphere_center_of_gravity(radius)<\/span><\/span>\n<\/span>\n    <\/span>print<\/span>(<\/span>f<\/span>"Area of the sphere: <\/span>{<\/span>area<\/span>}<\/span>"<\/span>)<\/span><\/span>\n    <\/span>print<\/span>(<\/span>f<\/span>"Surface area of the sphere: <\/span>{<\/span>surface_area<\/span>}<\/span>"<\/span>)<\/span><\/span>\n    <\/span>print<\/span>(<\/span>f<\/span>"Volume of the sphere: <\/span>{<\/span>volume<\/span>}<\/span>"<\/span>)<\/span><\/span>\n    <\/span>print<\/span>(<\/span>f<\/span>"Center of gravity of the sphere: <\/span>{<\/span>center_of_gravity<\/span>}<\/span>"<\/span>)<\/span><\/span>\n<\/span>\nmain()<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output Sample <\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Enter the radius of the sphere:  <\/span>6<\/span><\/span>\n<\/span>\nArea of the sphere: <\/span>452.3893421169302<\/span><\/span>\nSurface area of the sphere: <\/span>452.3893421169302<\/span><\/span>\nVolume of the sphere: <\/span>904.7786842338603<\/span><\/span>\nCenter of gravity of the sphere: <\/span>2.25<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    Additional code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\nfrom<\/span> mpl_toolkits.mplot3d <\/span>import<\/span> Axes3D<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>plot_wireframe_sphere<\/span>(<\/span>radius<\/span>):<\/span><\/span>\n    fig <\/span>=<\/span> plt.figure()<\/span><\/span>\n    ax <\/span>=<\/span> fig.add_subplot(<\/span>111<\/span>, <\/span>projection<\/span>=<\/span>'<\/span>3d<\/span>'<\/span>)<\/span><\/span>\n<\/span>\n    u <\/span>=<\/span> np.linspace(<\/span>0<\/span>, <\/span>2<\/span> <\/span>*<\/span> np.pi, <\/span>100<\/span>)<\/span><\/span>\n    v <\/span>=<\/span> np.linspace(<\/span>0<\/span>, np.pi, <\/span>100<\/span>)<\/span><\/span>\n    x <\/span>=<\/span> radius <\/span>*<\/span> np.outer(np.cos(u), np.sin(v))<\/span><\/span>\n    y <\/span>=<\/span> radius <\/span>*<\/span> np.outer(np.sin(u), np.sin(v))<\/span><\/span>\n    z <\/span>=<\/span> radius <\/span>*<\/span> np.outer(np.ones(np.size(u)), np.cos(v))<\/span><\/span>\n<\/span>\n    ax.plot_wireframe(x, y, z, <\/span>color<\/span>=<\/span>'<\/span>b<\/span>'<\/span>)<\/span><\/span>\n<\/span>\n    ax.set_xlabel(<\/span>'<\/span>X<\/span>'<\/span>)<\/span><\/span>\n    ax.set_ylabel(<\/span>'<\/span>Y<\/span>'<\/span>)<\/span><\/span>\n    ax.set_zlabel(<\/span>'<\/span>Z<\/span>'<\/span>)<\/span><\/span>\n    ax.set_title(<\/span>'<\/span>Wireframe Sphere<\/span>'<\/span>)<\/span><\/span>\n<\/span>\n    plt.show()<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>main<\/span>():<\/span><\/span>\n    radius <\/span>=<\/span> <\/span>float<\/span>(<\/span>input<\/span>(<\/span>"<\/span>Enter the radius of the sphere: <\/span>"<\/span>))<\/span><\/span>\n    plot_wireframe_sphere(radius)<\/span><\/span>\n<\/span>\nmain()<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    Enter the radius of the sphere: 6<\/p>\n\n\n\n
    \n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    \n\n\n\n
    2) Evaluation of improper integrals<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    from<\/span> scipy <\/span>import<\/span> integrate<\/span><\/span>\n<\/span>\n# Define the function to be integrated<\/span><\/span>\ndef<\/span> <\/span>integrand<\/span>(<\/span>x<\/span>):<\/span><\/span>\n    <\/span>return<\/span> (x<\/span>**<\/span>3<\/span> <\/span>+<\/span> <\/span>2<\/span>*<\/span>x <\/span>+<\/span> <\/span>7<\/span>) <\/span>\/<\/span>(x <\/span>+<\/span> <\/span>2<\/span>)<\/span><\/span>\n<\/span>\n# Define the limits of integration<\/span><\/span>\na <\/span>=<\/span> <\/span>0<\/span><\/span>\nb <\/span>=<\/span> <\/span>1<\/span><\/span>\n<\/span>\n# Evaluate the improper integral with increased limit<\/span><\/span>\nresult, error <\/span>=<\/span> integrate.quad(integrand, a, b)<\/span><\/span>\n<\/span>\nprint<\/span>(<\/span>f<\/span>"Result of the improper integral: <\/span>{<\/span>result<\/span>}<\/span>"<\/span>)<\/span><\/span>\nprint<\/span>(<\/span>f<\/span>"Estimated error: <\/span>{<\/span>error<\/span>}<\/span>"<\/span>)<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Result<\/span> <\/span>of<\/span> <\/span>the<\/span> <\/span>improper<\/span> integral<\/span>:<\/span> <\/span>3.3060077927925113<\/span><\/span>\nEstimated<\/span> error<\/span>:<\/span> <\/span>3.6704059711484397e-14<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    3) Finding gradient, divergent, curl and their geometrical interpretation<\/h2>\n\n\n\n
    Python Program<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> sympy <\/span>as<\/span> sp<\/span><\/span>\n<\/span>\n# Define symbols<\/span><\/span>\nx, y, z <\/span>=<\/span> sp.symbols(<\/span>'<\/span>x y z<\/span>'<\/span>)<\/span><\/span>\n<\/span>\n# Define vector field<\/span><\/span>\nF <\/span>=<\/span> sp.Matrix([x<\/span>**<\/span>2<\/span>, y<\/span>**<\/span>2<\/span>, z<\/span>**<\/span>2<\/span>])<\/span><\/span>\n<\/span>\n# Define variables for differentiation<\/span><\/span>\nvariables <\/span>=<\/span> (x, y, z)<\/span><\/span>\n<\/span>\n# Define gradient<\/span><\/span>\ngradient_F <\/span>=<\/span> sp.Matrix([sp.diff(F[i], var) <\/span>for<\/span> var <\/span>in<\/span> variables <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>3<\/span>)]).reshape(<\/span>3<\/span>, <\/span>3<\/span>)<\/span><\/span>\n<\/span>\n# Calculate divergence<\/span><\/span>\ndivergence_F <\/span>=<\/span> sp.diff(F[<\/span>0<\/span>], x) <\/span>+<\/span> sp.diff(F[<\/span>1<\/span>], y) <\/span>+<\/span> sp.diff(F[<\/span>2<\/span>], z)<\/span><\/span>\n<\/span>\n# Calculate curl<\/span><\/span>\ncurl_F <\/span>=<\/span> sp.Matrix([sp.diff(F[(i<\/span>+<\/span>1<\/span>) <\/span>%<\/span> <\/span>3<\/span>], variables[(i<\/span>+<\/span>2<\/span>) <\/span>%<\/span> <\/span>3<\/span>]) <\/span>-<\/span> sp.diff(F[(i<\/span>+<\/span>2<\/span>) <\/span>%<\/span> <\/span>3<\/span>], variables[(i<\/span>+<\/span>1<\/span>) <\/span>%<\/span> <\/span>3<\/span>])<\/span><\/span>\n                    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>3<\/span>)])<\/span><\/span>\n<\/span>\n# Print results<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Vector Field F(x, y, z) = <\/span>"<\/span>, F)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>\\n<\/span>Gradient of F:<\/span>"<\/span>)<\/span><\/span>\nprint<\/span>(gradient_F)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>\\n<\/span>Divergence of F:<\/span>"<\/span>)<\/span><\/span>\nprint<\/span>(divergence_F)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>\\n<\/span>Curl of F:<\/span>"<\/span>)<\/span><\/span>\nprint<\/span>(curl_F)<\/span><\/span>\n<\/span>\n# Geometrical interpretations<\/span><\/span>\n# Gradient gives the direction of maximum change in the scalar field.<\/span><\/span>\n# Divergence represents how much a vector field is spreading out or converging at a point.<\/span><\/span>\n# Curl indicates the rotation or "whirl" of the vector field at a point.<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Vector Field F(x, y, z) <\/span>=<\/span>  Matrix([[x<\/span>**<\/span>2<\/span>], [y<\/span>**<\/span>2<\/span>], [z<\/span>**<\/span>2<\/span>]])<\/span><\/span>\n<\/span>\nGradient of F:<\/span><\/span>\nMatrix([[<\/span>2<\/span>*<\/span>x, <\/span>0<\/span>, <\/span>0<\/span>], [<\/span>0<\/span>, <\/span>2<\/span>*<\/span>y, <\/span>0<\/span>], [<\/span>0<\/span>, <\/span>0<\/span>, <\/span>2<\/span>*<\/span>z]])<\/span><\/span>\n<\/span>\nDivergence of F:<\/span><\/span>\n2<\/span>*<\/span>x <\/span>+<\/span> <\/span>2<\/span>*<\/span>y <\/span>+<\/span> <\/span>2<\/span>*<\/span>z<\/span><\/span>\n<\/span>\nCurl of F:<\/span><\/span>\nMatrix([[<\/span>0<\/span>], [<\/span>0<\/span>], [<\/span>0<\/span>]])<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    \n\n\n\n
    4)Computation of basis and dimension for a vector space and Graphical representation of linear transformation<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>basis_and_dimension<\/span>(<\/span>matrix<\/span>):<\/span><\/span>\n    <\/span># Compute the reduced row echelon form<\/span><\/span>\n    rref_matrix <\/span>=<\/span> np.linalg.matrix_rank(matrix)<\/span><\/span>\n    dimension <\/span>=<\/span> np.linalg.matrix_rank(matrix.T)<\/span><\/span>\n    <\/span><\/span>\n    <\/span>return<\/span> rref_matrix, dimension<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>plot_linear_transformation<\/span>(<\/span>matrix<\/span>):<\/span><\/span>\n    <\/span># Create a set of 2D vectors<\/span><\/span>\n    vectors <\/span>=<\/span> np.array([[<\/span>1<\/span>, <\/span>0<\/span>], [<\/span>0<\/span>, <\/span>1<\/span>]])<\/span><\/span>\n    <\/span><\/span>\n    <\/span># Apply linear transformation to the vectors<\/span><\/span>\n    transformed_vectors <\/span>=<\/span> np.dot(matrix, vectors)<\/span><\/span>\n    <\/span><\/span>\n    <\/span># Plot original and transformed vectors<\/span><\/span>\n    plt.figure(<\/span>figsize<\/span>=<\/span>(<\/span>8<\/span>, <\/span>6<\/span>))<\/span><\/span>\n    plt.quiver([<\/span>0<\/span>, <\/span>0<\/span>], [<\/span>0<\/span>, <\/span>0<\/span>], vectors[<\/span>:<\/span>,<\/span>0<\/span>], vectors[<\/span>:<\/span>,<\/span>1<\/span>], <\/span>angles<\/span>=<\/span>'<\/span>xy<\/span>'<\/span>, <\/span>scale_units<\/span>=<\/span>'<\/span>xy<\/span>'<\/span>, <\/span>scale<\/span>=<\/span>1<\/span>, <\/span>color<\/span>=<\/span>[<\/span>'<\/span>r<\/span>'<\/span>, <\/span>'<\/span>b<\/span>'<\/span>], <\/span>label<\/span>=<\/span>'<\/span>Original Vectors<\/span>'<\/span>)<\/span><\/span>\n    plt.quiver([<\/span>0<\/span>, <\/span>0<\/span>], [<\/span>0<\/span>, <\/span>0<\/span>], transformed_vectors[<\/span>:<\/span>,<\/span>0<\/span>], transformed_vectors[<\/span>:<\/span>,<\/span>1<\/span>], <\/span>angles<\/span>=<\/span>'<\/span>xy<\/span>'<\/span>, <\/span>scale_units<\/span>=<\/span>'<\/span>xy<\/span>'<\/span>, <\/span>scale<\/span>=<\/span>1<\/span>, <\/span>color<\/span>=<\/span>[<\/span>'<\/span>g<\/span>'<\/span>, <\/span>'<\/span>y<\/span>'<\/span>], <\/span>label<\/span>=<\/span>'<\/span>Transformed Vectors<\/span>'<\/span>)<\/span><\/span>\n    plt.xlabel(<\/span>'<\/span>x<\/span>'<\/span>)<\/span><\/span>\n    plt.ylabel(<\/span>'<\/span>y<\/span>'<\/span>)<\/span><\/span>\n    plt.axhline(<\/span>0<\/span>, <\/span>color<\/span>=<\/span>'<\/span>black<\/span>'<\/span>,<\/span>linewidth<\/span>=<\/span>0.5<\/span>)<\/span><\/span>\n    plt.axvline(<\/span>0<\/span>, <\/span>color<\/span>=<\/span>'<\/span>black<\/span>'<\/span>,<\/span>linewidth<\/span>=<\/span>0.5<\/span>)<\/span><\/span>\n    plt.grid(<\/span>color<\/span> <\/span>=<\/span> <\/span>'<\/span>gray<\/span>'<\/span>, <\/span>linestyle<\/span> <\/span>=<\/span> <\/span>'<\/span>--<\/span>'<\/span>, <\/span>linewidth<\/span> <\/span>=<\/span> <\/span>0.5<\/span>)<\/span><\/span>\n    plt.axis(<\/span>'<\/span>equal<\/span>'<\/span>)<\/span><\/span>\n    plt.legend()<\/span><\/span>\n    plt.title(<\/span>'<\/span>Linear Transformation<\/span>'<\/span>)<\/span><\/span>\n    plt.show()<\/span><\/span>\n<\/span>\n# Example matrix representing a linear transformation<\/span><\/span>\nA <\/span>=<\/span> np.array([[<\/span>1<\/span>, <\/span>2<\/span>], [<\/span>3<\/span>, <\/span>4<\/span>]])<\/span><\/span>\n<\/span>\n# Compute basis and dimension<\/span><\/span>\nrref, dimension <\/span>=<\/span> basis_and_dimension(A)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Reduced Row Echelon Form:<\/span>\\n<\/span>"<\/span>, rref)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Dimension of Column Space:<\/span>"<\/span>, dimension)<\/span><\/span>\n<\/span>\n# Plot linear transformation<\/span><\/span>\nplot_linear_transformation(A)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Reduced<\/span> <\/span>Row<\/span> <\/span>Echelon<\/span> Form<\/span>:<\/span><\/span>\n <\/span>2<\/span><\/span>\nDimension<\/span> <\/span>of<\/span> <\/span>Column<\/span> Space<\/span>:<\/span> <\/span>2<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    5) Computing the inner product and orthogonality<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>inner_product<\/span>(<\/span>vector1<\/span>, <\/span>vector2<\/span>):<\/span><\/span>\n    <\/span># Compute the inner product (dot product) between two vectors<\/span><\/span>\n    <\/span>return<\/span> np.dot(vector1, vector2)<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>check_orthogonality<\/span>(<\/span>vector1<\/span>, <\/span>vector2<\/span>):<\/span><\/span>\n    <\/span># Check if two vectors are orthogonal (their inner product is zero)<\/span><\/span>\n    <\/span>return<\/span> inner_product(vector1, vector2) <\/span>==<\/span> <\/span>0<\/span><\/span>\n<\/span>\n# Example vectors<\/span><\/span>\nvector1 <\/span>=<\/span> np.array([<\/span>1<\/span>, <\/span>2<\/span>, <\/span>3<\/span>])<\/span><\/span>\nvector2 <\/span>=<\/span> np.array([<\/span>-<\/span>1<\/span>, <\/span>1<\/span>, <\/span>0<\/span>])<\/span><\/span>\n<\/span>\n# Compute the inner product of the vectors<\/span><\/span>\nresult <\/span>=<\/span> inner_product(vector1, vector2)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Inner product of vector1 and vector2:<\/span>"<\/span>, result)<\/span><\/span>\n<\/span>\n# Check orthogonality<\/span><\/span>\nif<\/span> check_orthogonality(vector1, vector2):<\/span><\/span>\n    <\/span>print<\/span>(<\/span>"<\/span>Vectors vector1 and vector2 are orthogonal.<\/span>"<\/span>)<\/span><\/span>\nelse<\/span>:<\/span><\/span>\n    <\/span>print<\/span>(<\/span>"<\/span>Vectors vector1 and vector2 are not orthogonal.<\/span>"<\/span>)<\/span><\/span>\n    <\/span><\/span>\n# Example vectors<\/span><\/span>\nvector3 <\/span>=<\/span> np.array([<\/span>1<\/span>, <\/span>2<\/span>, <\/span>3<\/span>])<\/span><\/span>\nvector4 <\/span>=<\/span> np.array([<\/span>-<\/span>1<\/span>, <\/span>1<\/span>, <\/span>2<\/span>])<\/span><\/span>\n<\/span>\n# Compute the inner product of the vectors<\/span><\/span>\nresult <\/span>=<\/span> inner_product(vector3, vector4)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Inner product of vector3 and vector4:<\/span>"<\/span>, result)<\/span><\/span>\n<\/span>\n# Check orthogonality<\/span><\/span>\nif<\/span> check_orthogonality(vector3, vector4):<\/span><\/span>\n    <\/span>print<\/span>(<\/span>"<\/span>Vectors vector3 and vector4 are orthogonal.<\/span>"<\/span>)<\/span><\/span>\nelse<\/span>:<\/span><\/span>\n    <\/span>print<\/span>(<\/span>"<\/span>Vectors vector3 and vector4 are not orthogonal.<\/span>"<\/span>)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Inner product of vector1 <\/span>and<\/span> vector2: <\/span>1<\/span><\/span>\nVectors vector1 <\/span>and<\/span> vector2 are <\/span>not<\/span> orthogonal.<\/span><\/span>\n<\/span>\nInner product of vector3 <\/span>and<\/span> vector4: <\/span>7<\/span><\/span>\nVectors vector3 <\/span>and<\/span> vector4 are <\/span>not<\/span> orthogonal.<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    \n\n\n\n
    6) Solution of algebraic and transcendental equations by Ramanujan\u2019s, Regula-Falsi and Newton-Raphson method<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>ramanujan<\/span>(<\/span>f<\/span>, <\/span>a<\/span>, <\/span>b<\/span>, <\/span>tol<\/span>=<\/span>1e-6<\/span>, <\/span>max_iter<\/span>=<\/span>100<\/span>):<\/span><\/span>\n    <\/span># Ramanujan's method for solving equations<\/span><\/span>\n    <\/span>for<\/span> _ <\/span>in<\/span> <\/span>range<\/span>(max_iter):<\/span><\/span>\n        c <\/span>=<\/span> (a <\/span>*<\/span> f(b) <\/span>-<\/span> b <\/span>*<\/span> f(a)) <\/span>\/<\/span> (f(b) <\/span>-<\/span> f(a))<\/span><\/span>\n        <\/span>if<\/span> np.abs(f(c)) <\/span><<\/span> tol:<\/span><\/span>\n            <\/span>return<\/span> c<\/span><\/span>\n        <\/span>if<\/span> f(a) <\/span>*<\/span> f(c) <\/span><<\/span> <\/span>0<\/span>:<\/span><\/span>\n            b <\/span>=<\/span> c<\/span><\/span>\n        <\/span>else<\/span>:<\/span><\/span>\n            a <\/span>=<\/span> c<\/span><\/span>\n    <\/span>raise<\/span> <\/span>ValueError<\/span>(<\/span>"<\/span>Ramanujan's method did not converge<\/span>"<\/span>)<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>regula_falsi<\/span>(<\/span>f<\/span>, <\/span>a<\/span>, <\/span>b<\/span>, <\/span>tol<\/span>=<\/span>1e-6<\/span>, <\/span>max_iter<\/span>=<\/span>100<\/span>):<\/span><\/span>\n    <\/span># Regula-Falsi method for solving equations<\/span><\/span>\n    <\/span>for<\/span> _ <\/span>in<\/span> <\/span>range<\/span>(max_iter):<\/span><\/span>\n        c <\/span>=<\/span> (a <\/span>*<\/span> f(b) <\/span>-<\/span> b <\/span>*<\/span> f(a)) <\/span>\/<\/span> (f(b) <\/span>-<\/span> f(a))<\/span><\/span>\n        <\/span>if<\/span> np.abs(f(c)) <\/span><<\/span> tol:<\/span><\/span>\n            <\/span>return<\/span> c<\/span><\/span>\n        <\/span>if<\/span> f(a) <\/span>*<\/span> f(c) <\/span><<\/span> <\/span>0<\/span>:<\/span><\/span>\n            b <\/span>=<\/span> c<\/span><\/span>\n        <\/span>else<\/span>:<\/span><\/span>\n            a <\/span>=<\/span> c<\/span><\/span>\n    <\/span>raise<\/span> <\/span>ValueError<\/span>(<\/span>"<\/span>Regula-Falsi method did not converge<\/span>"<\/span>)<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>newton_raphson<\/span>(<\/span>f<\/span>, <\/span>f_prime<\/span>, <\/span>x0<\/span>, <\/span>tol<\/span>=<\/span>1e-6<\/span>, <\/span>max_iter<\/span>=<\/span>100<\/span>):<\/span><\/span>\n    <\/span># Newton-Raphson method for solving equations<\/span><\/span>\n    x <\/span>=<\/span> x0<\/span><\/span>\n    <\/span>for<\/span> _ <\/span>in<\/span> <\/span>range<\/span>(max_iter):<\/span><\/span>\n        x_new <\/span>=<\/span> x <\/span>-<\/span> f(x) <\/span>\/<\/span> f_prime(x)<\/span><\/span>\n        <\/span>if<\/span> np.abs(f(x_new)) <\/span><<\/span> tol:<\/span><\/span>\n            <\/span>return<\/span> x_new<\/span><\/span>\n        x <\/span>=<\/span> x_new<\/span><\/span>\n    <\/span>raise<\/span> <\/span>ValueError<\/span>(<\/span>"<\/span>Newton-Raphson method did not converge<\/span>"<\/span>)<\/span><\/span>\n<\/span>\n# Example usage:<\/span><\/span>\n<\/span>\n# Define the function<\/span><\/span>\ndef<\/span> <\/span>f<\/span>(<\/span>x<\/span>):<\/span><\/span>\n    <\/span>return<\/span> x<\/span>**<\/span>3<\/span> <\/span>-<\/span> <\/span>2<\/span>*<\/span>x <\/span>-<\/span> <\/span>5<\/span><\/span>\n<\/span>\n# Define the derivative of the function<\/span><\/span>\ndef<\/span> <\/span>f_prime<\/span>(<\/span>x<\/span>):<\/span><\/span>\n    <\/span>return<\/span> <\/span>3<\/span>*<\/span>x<\/span>**<\/span>2<\/span> <\/span>-<\/span> <\/span>2<\/span><\/span>\n<\/span>\n# Initial guesses<\/span><\/span>\nx0_ramanujan <\/span>=<\/span> <\/span>1<\/span><\/span>\nx0_regula_falsi <\/span>=<\/span> <\/span>1<\/span><\/span>\nx0_newton_raphson <\/span>=<\/span> <\/span>1<\/span><\/span>\n<\/span>\n# Solve using Ramanujan's method<\/span><\/span>\nsolution_ramanujan <\/span>=<\/span> ramanujan(f, <\/span>1<\/span>, <\/span>3<\/span>)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Solution using Ramanujan's method:<\/span>"<\/span>, solution_ramanujan)<\/span><\/span>\n<\/span>\n# Solve using Regula-Falsi method<\/span><\/span>\nsolution_regula_falsi <\/span>=<\/span> regula_falsi(f, <\/span>1<\/span>, <\/span>3<\/span>)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Solution using Regula-Falsi method:<\/span>"<\/span>, solution_regula_falsi)<\/span><\/span>\n<\/span>\n# Solve using Newton-Raphson method<\/span><\/span>\nsolution_newton_raphson <\/span>=<\/span> newton_raphson(f, f_prime, <\/span>1<\/span>)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Solution using Newton-Raphson method:<\/span>"<\/span>, solution_newton_raphson)<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    Solution<\/span> <\/span>using<\/span> <\/span>Ramanujan<\/span>'<\/span>s method: 2.09455140006329<\/span>8<\/span><\/span>\nSolution<\/span> <\/span>using<\/span> <\/span>Regula<\/span>-Falsi method<\/span>:<\/span> <\/span>2.094551400063298<\/span><\/span>\nSolution<\/span> using Newton-Raphson method<\/span>:<\/span> <\/span>2.0945514815642077<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    \n\n\n\n
    7) Interpolation\/Extrapolation using Newton\u2019s forward and backward difference formula<\/h2>\n\n\n\n
    Python Program<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>forward_difference_coefficients<\/span>(<\/span>y<\/span>):<\/span><\/span>\n    n <\/span>=<\/span> <\/span>len<\/span>(y)<\/span><\/span>\n    coeffs <\/span>=<\/span> np.zeros((n, n))<\/span><\/span>\n    coeffs[<\/span>:<\/span>, <\/span>0<\/span>] <\/span>=<\/span> y<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> j <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, n):<\/span><\/span>\n        <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(n <\/span>-<\/span> j):<\/span><\/span>\n            coeffs[i, j] <\/span>=<\/span> coeffs[i<\/span>+<\/span>1<\/span>, j<\/span>-<\/span>1<\/span>] <\/span>-<\/span> coeffs[i, j<\/span>-<\/span>1<\/span>]<\/span><\/span>\n            <\/span><\/span>\n    <\/span>return<\/span> coeffs<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>backward_difference_coefficients<\/span>(<\/span>y<\/span>):<\/span><\/span>\n    n <\/span>=<\/span> <\/span>len<\/span>(y)<\/span><\/span>\n    coeffs <\/span>=<\/span> np.zeros((n, n))<\/span><\/span>\n    coeffs[<\/span>:<\/span>, <\/span>0<\/span>] <\/span>=<\/span> y<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> j <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, n):<\/span><\/span>\n        <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(j, n):<\/span><\/span>\n            coeffs[i, j] <\/span>=<\/span> coeffs[i, j<\/span>-<\/span>1<\/span>] <\/span>-<\/span> coeffs[i<\/span>-<\/span>1<\/span>, j<\/span>-<\/span>1<\/span>]<\/span><\/span>\n            <\/span><\/span>\n    <\/span>return<\/span> coeffs<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>newton_interpolation<\/span>(<\/span>x<\/span>, <\/span>y<\/span>, <\/span>coeffs<\/span>, <\/span>xi<\/span>):<\/span><\/span>\n    n <\/span>=<\/span> <\/span>len<\/span>(x)<\/span><\/span>\n    h <\/span>=<\/span> x[<\/span>1<\/span>] <\/span>-<\/span> x[<\/span>0<\/span>]<\/span><\/span>\n    <\/span><\/span>\n    fxi <\/span>=<\/span> coeffs[<\/span>0<\/span>, <\/span>0<\/span>]<\/span><\/span>\n    <\/span>for<\/span> j <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, n):<\/span><\/span>\n        term <\/span>=<\/span> coeffs[<\/span>0<\/span>, j]<\/span><\/span>\n        <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(j):<\/span><\/span>\n            term <\/span>*=<\/span> (xi <\/span>-<\/span> x[i]) <\/span>\/<\/span> ((j<\/span>+<\/span>1<\/span>) <\/span>*<\/span> h)<\/span><\/span>\n        fxi <\/span>+=<\/span> term<\/span><\/span>\n        <\/span><\/span>\n    <\/span>return<\/span> fxi<\/span><\/span>\n<\/span>\n# Data points<\/span><\/span>\nx <\/span>=<\/span> np.array([<\/span>0<\/span>, <\/span>1<\/span>, <\/span>2<\/span>, <\/span>3<\/span>, <\/span>4<\/span>])<\/span><\/span>\ny <\/span>=<\/span> np.array([<\/span>1<\/span>, <\/span>2<\/span>, <\/span>4<\/span>, <\/span>8<\/span>, <\/span>16<\/span>])<\/span><\/span>\n<\/span>\n# Coefficients for forward and backward difference formula<\/span><\/span>\ncoeffs_forward <\/span>=<\/span> forward_difference_coefficients(y)<\/span><\/span>\ncoeffs_backward <\/span>=<\/span> backward_difference_coefficients(y)<\/span><\/span>\n<\/span>\n# Visualization of coefficients<\/span><\/span>\nplt.figure(<\/span>figsize<\/span>=<\/span>(<\/span>12<\/span>, <\/span>6<\/span>))<\/span><\/span>\n<\/span>\n# Forward Difference Coefficients<\/span><\/span>\nplt.subplot(<\/span>1<\/span>, <\/span>2<\/span>, <\/span>1<\/span>)<\/span><\/span>\nplt.imshow(coeffs_forward, <\/span>cmap<\/span>=<\/span>'<\/span>coolwarm<\/span>'<\/span>, <\/span>aspect<\/span>=<\/span>'<\/span>auto<\/span>'<\/span>)<\/span><\/span>\nplt.title(<\/span>'<\/span>Forward Difference Coefficients<\/span>'<\/span>)<\/span><\/span>\nplt.xlabel(<\/span>'<\/span>Order of Difference<\/span>'<\/span>)<\/span><\/span>\nplt.ylabel(<\/span>'<\/span>Data Points<\/span>'<\/span>)<\/span><\/span>\nplt.colorbar(<\/span>label<\/span>=<\/span>'<\/span>Coefficient Value<\/span>'<\/span>)<\/span><\/span>\n<\/span>\n# Backward Difference Coefficients<\/span><\/span>\nplt.subplot(<\/span>1<\/span>, <\/span>2<\/span>, <\/span>2<\/span>)<\/span><\/span>\nplt.imshow(coeffs_backward, <\/span>cmap<\/span>=<\/span>'<\/span>coolwarm<\/span>'<\/span>, <\/span>aspect<\/span>=<\/span>'<\/span>auto<\/span>'<\/span>)<\/span><\/span>\nplt.title(<\/span>'<\/span>Backward Difference Coefficients<\/span>'<\/span>)<\/span><\/span>\nplt.xlabel(<\/span>'<\/span>Order of Difference<\/span>'<\/span>)<\/span><\/span>\nplt.ylabel(<\/span>'<\/span>Data Points<\/span>'<\/span>)<\/span><\/span>\nplt.colorbar(<\/span>label<\/span>=<\/span>'<\/span>Coefficient Value<\/span>'<\/span>)<\/span><\/span>\n<\/span>\nplt.tight_layout()<\/span><\/span>\nplt.show()<\/span><\/span>\n<\/span>\n# Interpolation<\/span><\/span>\nxi <\/span>=<\/span> <\/span>2.5<\/span><\/span>\nforward_interpolation <\/span>=<\/span> newton_interpolation(x, y, coeffs_forward, xi)<\/span><\/span>\nbackward_interpolation <\/span>=<\/span> newton_interpolation(x, y, coeffs_backward, xi)<\/span><\/span>\n<\/span>\nprint<\/span>(<\/span>"<\/span>Interpolated value at xi using forward difference formula:<\/span>"<\/span>, forward_interpolation)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Interpolated value at xi using backward difference formula:<\/span>"<\/span>, backward_interpolation)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    \n\n\n\n
    8) Computation of area under the curve using Trapezoidal, Simpson\u2019s (1\/3)rd<\/sup> and (3\/8)th<\/sup> rule<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>trapezoidal_rule<\/span>(<\/span>f<\/span>, <\/span>a<\/span>, <\/span>b<\/span>, <\/span>n<\/span>):<\/span><\/span>\n    h <\/span>=<\/span> (b <\/span>-<\/span> a) <\/span>\/<\/span> n<\/span><\/span>\n    x <\/span>=<\/span> np.linspace(a, b, n<\/span>+<\/span>1<\/span>)<\/span><\/span>\n    y <\/span>=<\/span> f(x)<\/span><\/span>\n    area <\/span>=<\/span> h <\/span>*<\/span> (np.sum(y) <\/span>-<\/span> <\/span>0.5<\/span> <\/span>*<\/span> (y[<\/span>0<\/span>] <\/span>+<\/span> y[<\/span>-<\/span>1<\/span>]))<\/span><\/span>\n    <\/span>return<\/span> area, x, y<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>simpsons_one_third_rule<\/span>(<\/span>f<\/span>, <\/span>a<\/span>, <\/span>b<\/span>, <\/span>n<\/span>):<\/span><\/span>\n    h <\/span>=<\/span> (b <\/span>-<\/span> a) <\/span>\/<\/span> n<\/span><\/span>\n    x <\/span>=<\/span> np.linspace(a, b, n<\/span>+<\/span>1<\/span>)<\/span><\/span>\n    y <\/span>=<\/span> f(x)<\/span><\/span>\n    area <\/span>=<\/span> h<\/span>\/<\/span>3<\/span> <\/span>*<\/span> (y[<\/span>0<\/span>] <\/span>+<\/span> <\/span>4<\/span>*<\/span>np.sum(y[<\/span>1<\/span>:-<\/span>1<\/span>:<\/span>2<\/span>]) <\/span>+<\/span> <\/span>2<\/span>*<\/span>np.sum(y[<\/span>2<\/span>:-<\/span>2<\/span>:<\/span>2<\/span>]) <\/span>+<\/span> y[<\/span>-<\/span>1<\/span>])<\/span><\/span>\n    <\/span>return<\/span> area, x, y<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>simpsons_three_eighth_rule<\/span>(<\/span>f<\/span>, <\/span>a<\/span>, <\/span>b<\/span>, <\/span>n<\/span>):<\/span><\/span>\n    h <\/span>=<\/span> (b <\/span>-<\/span> a) <\/span>\/<\/span> n<\/span><\/span>\n    x <\/span>=<\/span> np.linspace(a, b, n<\/span>+<\/span>1<\/span>)<\/span><\/span>\n    y <\/span>=<\/span> f(x)<\/span><\/span>\n    area <\/span>=<\/span> <\/span>3<\/span>*<\/span>h<\/span>\/<\/span>8<\/span> <\/span>*<\/span> (y[<\/span>0<\/span>] <\/span>+<\/span> <\/span>3<\/span>*<\/span>np.sum(y[<\/span>1<\/span>:-<\/span>1<\/span>:<\/span>3<\/span>]) <\/span>+<\/span> <\/span>3<\/span>*<\/span>np.sum(y[<\/span>2<\/span>:-<\/span>2<\/span>:<\/span>3<\/span>]) <\/span>+<\/span> <\/span>2<\/span>*<\/span>np.sum(y[<\/span>3<\/span>:-<\/span>3<\/span>:<\/span>3<\/span>]) <\/span>+<\/span> y[<\/span>-<\/span>1<\/span>])<\/span><\/span>\n    <\/span>return<\/span> area, x, y<\/span><\/span>\n<\/span>\n# Example function<\/span><\/span>\ndef<\/span> <\/span>f<\/span>(<\/span>x<\/span>):<\/span><\/span>\n    <\/span>return<\/span> np.sin(x)<\/span><\/span>\n<\/span>\n# Integration interval<\/span><\/span>\na <\/span>=<\/span> <\/span>0<\/span><\/span>\nb <\/span>=<\/span> np.pi<\/span><\/span>\n<\/span>\n# Number of subintervals<\/span><\/span>\nn <\/span>=<\/span> <\/span>10<\/span><\/span>\n<\/span>\n# Compute area using different rules<\/span><\/span>\narea_trapezoidal, x_trap, y_trap <\/span>=<\/span> trapezoidal_rule(f, a, b, n)<\/span><\/span>\narea_simpsons_one_third, x_simp13, y_simp13 <\/span>=<\/span> simpsons_one_third_rule(f, a, b, n)<\/span><\/span>\narea_simpsons_three_eighth, x_simp38, y_simp38 <\/span>=<\/span> simpsons_three_eighth_rule(f, a, b, n)<\/span><\/span>\n<\/span>\n# Plot the function<\/span><\/span>\nx <\/span>=<\/span> np.linspace(a, b, <\/span>100<\/span>)<\/span><\/span>\ny <\/span>=<\/span> f(x)<\/span><\/span>\n<\/span>\nplt.figure(<\/span>figsize<\/span>=<\/span>(<\/span>10<\/span>, <\/span>6<\/span>))<\/span><\/span>\nplt.plot(x, y, <\/span>label<\/span>=<\/span>'<\/span>Function: sin(x)<\/span>'<\/span>)<\/span><\/span>\nplt.fill_between(x_trap, <\/span>0<\/span>, y_trap, <\/span>alpha<\/span>=<\/span>0.3<\/span>, <\/span>label<\/span>=<\/span>'<\/span>Trapezoidal Rule Area<\/span>'<\/span>)<\/span><\/span>\nplt.fill_between(x_simp13, <\/span>0<\/span>, y_simp13, <\/span>alpha<\/span>=<\/span>0.3<\/span>, <\/span>label<\/span>=<\/span>"<\/span>Simpson's 1\/3 Rule Area<\/span>"<\/span>)<\/span><\/span>\nplt.fill_between(x_simp38, <\/span>0<\/span>, y_simp38, <\/span>alpha<\/span>=<\/span>0.3<\/span>, <\/span>label<\/span>=<\/span>"<\/span>Simpson's 3\/8 Rule Area<\/span>"<\/span>)<\/span><\/span>\nplt.xlabel(<\/span>'<\/span>x<\/span>'<\/span>)<\/span><\/span>\nplt.ylabel(<\/span>'<\/span>y<\/span>'<\/span>)<\/span><\/span>\nplt.title(<\/span>'<\/span>Area Under the Curve Approximation<\/span>'<\/span>)<\/span><\/span>\nplt.legend()<\/span><\/span>\nplt.grid(<\/span>True<\/span>)<\/span><\/span>\nplt.show()<\/span><\/span>\n<\/span>\n# Print results<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Area under the curve using Trapezoidal rule:<\/span>"<\/span>, area_trapezoidal)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Area under the curve using Simpson's 1\/3 rule:<\/span>"<\/span>, area_simpsons_one_third)<\/span><\/span>\nprint<\/span>(<\/span>"<\/span>Area under the curve using Simpson's 3\/8 rule:<\/span>"<\/span>, area_simpsons_three_eighth)<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    \n\n\n\n
    \n\n\n\n
    9) Solution of ODE of first order and first degree by Taylor\u2019s series and Modified Euler\u2019s method<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>taylors_series<\/span>(<\/span>f<\/span>, <\/span>y0<\/span>, <\/span>t0<\/span>, <\/span>tn<\/span>, <\/span>h<\/span>):<\/span><\/span>\n    t_values <\/span>=<\/span> np.arange(t0, tn<\/span>+<\/span>h, h)<\/span><\/span>\n    y_values <\/span>=<\/span> np.zeros_like(t_values)<\/span><\/span>\n    y_values[<\/span>0<\/span>] <\/span>=<\/span> y0<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, <\/span>len<\/span>(t_values)):<\/span><\/span>\n        t <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y <\/span>=<\/span> y_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y_prime <\/span>=<\/span> f(t, y)<\/span><\/span>\n        y_double_prime <\/span>=<\/span> f(t, y_prime)<\/span><\/span>\n        y_values[i] <\/span>=<\/span> y <\/span>+<\/span> h<\/span>*<\/span>y_prime <\/span>+<\/span> (h<\/span>**<\/span>2<\/span> <\/span>\/<\/span> <\/span>2<\/span>)<\/span>*<\/span>y_double_prime<\/span><\/span>\n    <\/span><\/span>\n    <\/span>return<\/span> t_values, y_values<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>modified_euler<\/span>(<\/span>f<\/span>, <\/span>y0<\/span>, <\/span>t0<\/span>, <\/span>tn<\/span>, <\/span>h<\/span>):<\/span><\/span>\n    t_values <\/span>=<\/span> np.arange(t0, tn<\/span>+<\/span>h, h)<\/span><\/span>\n    y_values <\/span>=<\/span> np.zeros_like(t_values)<\/span><\/span>\n    y_values[<\/span>0<\/span>] <\/span>=<\/span> y0<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, <\/span>len<\/span>(t_values)):<\/span><\/span>\n        t <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y <\/span>=<\/span> y_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y_prime <\/span>=<\/span> f(t, y)<\/span><\/span>\n        y_modified <\/span>=<\/span> y <\/span>+<\/span> h<\/span>*<\/span>y_prime<\/span><\/span>\n        y_prime_modified <\/span>=<\/span> f(t<\/span>+<\/span>h, y_modified)<\/span><\/span>\n        y_values[i] <\/span>=<\/span> y <\/span>+<\/span> (h<\/span>\/<\/span>2<\/span>)<\/span>*<\/span>(y_prime <\/span>+<\/span> y_prime_modified)<\/span><\/span>\n    <\/span><\/span>\n    <\/span>return<\/span> t_values, y_values<\/span><\/span>\n<\/span>\n# Example ODE: y' = y - t^2 + 1, y(0) = 0.5<\/span><\/span>\ndef<\/span> <\/span>f<\/span>(<\/span>t<\/span>, <\/span>y<\/span>):<\/span><\/span>\n    <\/span>return<\/span> y <\/span>-<\/span> t<\/span>**<\/span>2<\/span> <\/span>+<\/span> <\/span>1<\/span><\/span>\n<\/span>\n# Initial conditions<\/span><\/span>\ny0 <\/span>=<\/span> <\/span>0.5<\/span><\/span>\nt0 <\/span>=<\/span> <\/span>0<\/span><\/span>\ntn <\/span>=<\/span> <\/span>2<\/span><\/span>\nh <\/span>=<\/span> <\/span>0.2<\/span><\/span>\n<\/span>\n# Solve using Taylor's series method<\/span><\/span>\nt_values_taylor, y_values_taylor <\/span>=<\/span> taylors_series(f, y0, t0, tn, h)<\/span><\/span>\n<\/span>\n# Solve using Modified Euler's method<\/span><\/span>\nt_values_euler, y_values_euler <\/span>=<\/span> modified_euler(f, y0, t0, tn, h)<\/span><\/span>\n<\/span>\n# Plot the solutions<\/span><\/span>\nplt.figure(<\/span>figsize<\/span>=<\/span>(<\/span>10<\/span>, <\/span>6<\/span>))<\/span><\/span>\nplt.plot(t_values_taylor, y_values_taylor, <\/span>label<\/span>=<\/span>"<\/span>Taylor's Series<\/span>"<\/span>)<\/span><\/span>\nplt.plot(t_values_euler, y_values_euler, <\/span>label<\/span>=<\/span>"<\/span>Modified Euler's Method<\/span>"<\/span>)<\/span><\/span>\nplt.xlabel(<\/span>'<\/span>t<\/span>'<\/span>)<\/span><\/span>\nplt.ylabel(<\/span>'<\/span>y<\/span>'<\/span>)<\/span><\/span>\nplt.title(<\/span>'<\/span>Solution of ODE: y<\/span>\\'<\/span> = y - t^2 + 1<\/span>'<\/span>)<\/span><\/span>\nplt.legend()<\/span><\/span>\nplt.grid(<\/span>True<\/span>)<\/span><\/span>\nplt.show()<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    \n\n\n\n
    10) Solution of ODE of first order and first degree by Runge-Kutta 4th order and Milne\u2019s predictor-corrector method<\/h2>\n\n\n\n
    Python Code<\/h3>\n\n\n\n
    <\/circle><\/circle><\/circle><\/g><\/svg><\/span><\/path><\/path><\/svg><\/span>
    import<\/span> numpy <\/span>as<\/span> np<\/span><\/span>\nimport<\/span> matplotlib.pyplot <\/span>as<\/span> plt<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>runge_kutta_4th_order<\/span>(<\/span>f<\/span>, <\/span>y0<\/span>, <\/span>t0<\/span>, <\/span>tn<\/span>, <\/span>h<\/span>):<\/span><\/span>\n    t_values <\/span>=<\/span> np.arange(t0, tn<\/span>+<\/span>h, h)<\/span><\/span>\n    y_values <\/span>=<\/span> np.zeros_like(t_values)<\/span><\/span>\n    y_values[<\/span>0<\/span>] <\/span>=<\/span> y0<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, <\/span>len<\/span>(t_values)):<\/span><\/span>\n        t <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y <\/span>=<\/span> y_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        <\/span><\/span>\n        k1 <\/span>=<\/span> h <\/span>*<\/span> f(t, y)<\/span><\/span>\n        k2 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h<\/span>\/<\/span>2<\/span>, y <\/span>+<\/span> k1<\/span>\/<\/span>2<\/span>)<\/span><\/span>\n        k3 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h<\/span>\/<\/span>2<\/span>, y <\/span>+<\/span> k2<\/span>\/<\/span>2<\/span>)<\/span><\/span>\n        k4 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h, y <\/span>+<\/span> k3)<\/span><\/span>\n        <\/span><\/span>\n        y_values[i] <\/span>=<\/span> y <\/span>+<\/span> (k1 <\/span>+<\/span> <\/span>2<\/span>*<\/span>k2 <\/span>+<\/span> <\/span>2<\/span>*<\/span>k3 <\/span>+<\/span> k4) <\/span>\/<\/span> <\/span>6<\/span><\/span>\n    <\/span><\/span>\n    <\/span>return<\/span> t_values, y_values<\/span><\/span>\n<\/span>\ndef<\/span> <\/span>milnes_predictor_corrector<\/span>(<\/span>f<\/span>, <\/span>y0<\/span>, <\/span>t0<\/span>, <\/span>tn<\/span>, <\/span>h<\/span>):<\/span><\/span>\n    t_values <\/span>=<\/span> np.arange(t0, tn<\/span>+<\/span>h, h)<\/span><\/span>\n    y_values <\/span>=<\/span> np.zeros_like(t_values)<\/span><\/span>\n    y_values[<\/span>0<\/span>] <\/span>=<\/span> y0<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>1<\/span>, <\/span>4<\/span>):<\/span><\/span>\n        t <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y <\/span>=<\/span> y_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        k1 <\/span>=<\/span> h <\/span>*<\/span> f(t, y)<\/span><\/span>\n        k2 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h<\/span>\/<\/span>2<\/span>, y <\/span>+<\/span> k1<\/span>\/<\/span>2<\/span>)<\/span><\/span>\n        k3 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h<\/span>\/<\/span>2<\/span>, y <\/span>+<\/span> k2<\/span>\/<\/span>2<\/span>)<\/span><\/span>\n        k4 <\/span>=<\/span> h <\/span>*<\/span> f(t <\/span>+<\/span> h, y <\/span>+<\/span> k3)<\/span><\/span>\n        <\/span><\/span>\n        y_values[i] <\/span>=<\/span> y <\/span>+<\/span> (k1 <\/span>+<\/span> <\/span>2<\/span>*<\/span>k2 <\/span>+<\/span> <\/span>2<\/span>*<\/span>k3 <\/span>+<\/span> k4) <\/span>\/<\/span> <\/span>6<\/span><\/span>\n    <\/span><\/span>\n    <\/span>for<\/span> i <\/span>in<\/span> <\/span>range<\/span>(<\/span>3<\/span>, <\/span>len<\/span>(t_values)):<\/span><\/span>\n        t_n3 <\/span>=<\/span> t_values[i<\/span>-<\/span>3<\/span>]<\/span><\/span>\n        t_n2 <\/span>=<\/span> t_values[i<\/span>-<\/span>2<\/span>]<\/span><\/span>\n        t_n1 <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y_n3 <\/span>=<\/span> y_values[i<\/span>-<\/span>3<\/span>]<\/span><\/span>\n        y_n2 <\/span>=<\/span> y_values[i<\/span>-<\/span>2<\/span>]<\/span><\/span>\n        y_n1 <\/span>=<\/span> y_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        <\/span><\/span>\n        t <\/span>=<\/span> t_values[i<\/span>-<\/span>1<\/span>]<\/span><\/span>\n        y_p <\/span>=<\/span> y_n1 <\/span>+<\/span> <\/span>4<\/span>*<\/span>h<\/span>\/<\/span>3<\/span> <\/span>*<\/span> (<\/span>2<\/span>*<\/span>f(t, y_n1) <\/span>-<\/span> f(t_n1, y_n2) <\/span>+<\/span> <\/span>2<\/span>*<\/span>f(t_n2, y_n3))<\/span><\/span>\n        t <\/span>=<\/span> t_values[i]<\/span><\/span>\n        y_values[i] <\/span>=<\/span> y_n1 <\/span>+<\/span> h<\/span>\/<\/span>3<\/span> <\/span>*<\/span> (f(t, y_p) <\/span>+<\/span> <\/span>4<\/span>*<\/span>f(t_n1, y_n1) <\/span>+<\/span> f(t_n2, y_n2))<\/span><\/span>\n    <\/span><\/span>\n    <\/span>return<\/span> t_values, y_values<\/span><\/span>\n<\/span>\n# Example ODE: y' = y - t^2 + 1, y(0) = 0.5<\/span><\/span>\ndef<\/span> <\/span>f<\/span>(<\/span>t<\/span>, <\/span>y<\/span>):<\/span><\/span>\n    <\/span>return<\/span> y <\/span>-<\/span> t<\/span>**<\/span>2<\/span> <\/span>+<\/span> <\/span>1<\/span><\/span>\n<\/span>\n# Initial conditions<\/span><\/span>\ny0 <\/span>=<\/span> <\/span>0.5<\/span><\/span>\nt0 <\/span>=<\/span> <\/span>0<\/span><\/span>\ntn <\/span>=<\/span> <\/span>2<\/span><\/span>\nh <\/span>=<\/span> <\/span>0.2<\/span><\/span>\n<\/span>\n# Solve using Runge-Kutta 4th order method<\/span><\/span>\nt_values_rk4, y_values_rk4 <\/span>=<\/span> runge_kutta_4th_order(f, y0, t0, tn, h)<\/span><\/span>\n<\/span>\n# Solve using Milne's predictor-corrector method<\/span><\/span>\nt_values_milne, y_values_milne <\/span>=<\/span> milnes_predictor_corrector(f, y0, t0, tn, h)<\/span><\/span>\n<\/span>\n# Plot the solutions<\/span><\/span>\nplt.figure(<\/span>figsize<\/span>=<\/span>(<\/span>10<\/span>, <\/span>6<\/span>))<\/span><\/span>\nplt.plot(t_values_rk4, y_values_rk4, <\/span>label<\/span>=<\/span>"<\/span>Runge-Kutta 4th Order<\/span>"<\/span>)<\/span><\/span>\nplt.plot(t_values_milne, y_values_milne, <\/span>label<\/span>=<\/span>"<\/span>Milne's Predictor-Corrector<\/span>"<\/span>)<\/span><\/span>\nplt.xlabel(<\/span>'<\/span>t<\/span>'<\/span>)<\/span><\/span>\nplt.ylabel(<\/span>'<\/span>y<\/span>'<\/span>)<\/span><\/span>\nplt.title(<\/span>'<\/span>Solution of ODE: y<\/span>\\'<\/span> = y - t^2 + 1<\/span>'<\/span>)<\/span><\/span>\nplt.legend()<\/span><\/span>\nplt.grid(<\/span>True<\/span>)<\/span><\/span>\nplt.show()<\/span><\/span>\n<\/span><\/code><\/pre><\/div>\n\n\n\n
    To run this program online click n the link below<\/p>\n\n\n\n
    Output<\/h3>\n\n\n\n
    \"\"<\/a><\/figure>\n\n\n\n
    If you are also looking for other Lab Manuals, head over to my following blog :<\/p>\n\n\n\n
    \n
    VTU Lab Manuals using FOSS<\/a><\/blockquote>
    P10 Program<\/a><\/p>\n\n\n\n
    P09 Program<\/a><\/p>\n\n\n\n
    P08 Program<\/a><\/p>\n\n\n\n
    P07 Program<\/a><\/p>\n\n\n\n
    P06 Program<\/a><\/p>\n\n\n\n
    P05 Program<\/a><\/p>\n\n\n\n
    P04 Program<\/a><\/p>\n\n\n\n
    P03 Program<\/a><\/p>\n\n\n\n
    P02 Program<\/a><\/p>\n\n\n\n
    P01 Program<\/a><\/p>\n\n\n\n