norm
==============================================

Purpose
----------------

Computes one of several specified matrix norms, or a vector p-norm.

Format
----------------
.. function:: n = norm(A[, norm_type])

    :param A: Data.
    :type A: Nx1 vector or NxN matrix

    :param norm_type: Optional, specifying which norm to compute. 

        ========= ==================
        Matrix norm options
        ============================
        1         Scalar, equivalent to: ``maxc(sumc(abs(x)))``
        2         Scalar,  the spectral norm, equivalent to: ``maxc(svds(x))``
        \_\__INFP  Scalar, the infinity norm, equivalent to: ``maxc(sumr(abs(x)))``
        "fro"     String, the Frobenius norm, equivalent to: ``sqrt(sumc(vecr(x.^2)))``.
        "nuc"     String, the nuclear norm, equivalent to: ``sumc(svds(A))``.
        ========= ==================
   
        ========= ==================
        Vector norm options
        ============================
        1         Scalar, equivalent to: ``sumcs(abs(x))`` for column vectors, or ``sumr(abs(x))`` for row vectors.
        2         Scalar, the l2 or Euclidean norm, equivalent to: ``sqrt(sumc(x.^2))``, or ``sqrt(sumr(x.^2))``
        p         Scalar, any real number, equivalent to: ``sumc(abs(x.^p)).^(1/p)``, or ``sumr(abs(x.^p)).^(1/p)``
        \_\__INFP  Scalar, equivalent to: ``maxc(abs(x))``, or ``maxc(abs(x'))``
        \_\_INFN  Scalar, equivalent to: ``minc(abs(x))``, or ``minc(abs(x'))``
        ========= ==================

    :type norm_type: String or scalar

    :return n: The requested norm of *A*.

    :rtype n: Scalar

Examples
----------------

Matrix norms
++++++++++++

::

    // Create 4x3 matrix
    A = { 0.35148166       0.53337376      -0.91676553,
          0.89133334      0.099774011        1.1669254,
         -0.54380494      -0.52901019       0.38900312,
         -0.67434004       -1.1692513      -0.14388126 };
    
    // Matrix 1 norm
    n_1 = norm(A, 1);
    
    // Matrix spectral norm
    n_2 = norm(A, 2);
    
    // Matrix Infinity norm
    n_inf = norm(A, __INFP);
    
    // Matrix Frobenius norm
    n_fro = norm(A, "fro");
    
    // Matrix nuclear norm
    n_nuc = norm(A, "nuc");
    
    // Singular values of 'A'
    s = svds(A);

The above code will make the following assignments:

::

    n_1   = 2.6166     n_2   = 1.7835    n_inf = 2.1580
    n_fro = 2.4462     n_nuc = 3.8478
    
    s =   1.7835
          1.6121
          0.4522

Vector norms
++++++++++++

::

    // Column vector
    v = { 0.0502,
         -0.7841,
          0.5719,
         -0.8668 };
    
    // Vector 1 norm
    n_1 = norm(v, 1);
    
    // Vector Euclidean norm
    n_2 = norm(v, 2);
    
    // Vector p norm
    n_p = norm(v, 3);
    
    n_pos_inf = norm(v, __INFP);
    n_neg_inf = norm(v, __INFN);

The above code will make the following assignments:

::

    n_1       = 2.2730    n_2       = 1.3022    n_p = 1.0971
    n_pos_inf = 0.8668    n_neg_inf = 0.0502

::

    // Row vector
    vt = { -0.5396  -0.0972  -0.0176   1.0552 };
    
    // Vector 1 norm
    n_1 = norm(vt, 1);
    
    // Vector Euclidean norm
    n_2 = norm(vt, 2);
    
    // Vector p norm
    n_p = norm(vt, 3);
    
    n_pos_inf = norm(vt, __INFP);
    n_neg_inf = norm(vt, __INFN);

The above code will make the following assignments:

::

    n_1       = 1.7096    n_2       = 1.1893    n_p = 1.0005
    n_pos_inf = 1.0552    n_neg_inf = 0.0176

Remarks
-------

-  To compute the Euclidean norm of each column vector of a matrix,
   call:

   ::

      n = sqrt(dot(A, A));

.. seealso:: Functions :func:`detl`, :func:`dot`, :func:`rank`