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Norm

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Norm[expr]
    gives the 2-norm of a number, vector, or matrix (Frobenius norm for
    matrices).
Norm[expr, p]
    gives the p-norm: Abs[expr] for scalars; (Sum |xi|^p)^(1/p) for
    vectors with 1 <= p < Infinity; Max[Abs[expr]] for p == Infinity;
    induced operator norm for matrices when p is 1, 2, or Infinity.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Norm[{x, y, z}]
Out[1]= Sqrt[Abs[x]^2 + Abs[y]^2 + Abs[z]^2]

In[2]:= Norm[-2 + I]
Out[2]= Sqrt[5]

In[3]:= v = {1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1}; Norm[v]
Out[3]= Sqrt[5]

In[4]:= Norm[{x, y, z}, p]
Out[4]= (Abs[x]^p + Abs[y]^p + Abs[z]^p)^(1/p)

In[5]:= Norm[{x, y, z}, Infinity]
Out[5]= Max[Abs[x], Abs[y], Abs[z]]

In[6]:= Norm[{{a11, a12}, {a21, a22}}, "Frobenius"]
Out[6]= Sqrt[Abs[a11]^2 + Abs[a12]^2 + Abs[a21]^2 + Abs[a22]^2]

Implementation notes

Algorithm. builtin_norm uses get_tensor_dims to classify the argument and then builds a symbolic norm expression that it evaluates. A rank-0 scalar reduces to Abs[x] (the p argument is rejected for scalars). For a rank-1 vector (or any tensor when p is the string "Frobenius") it flattens to Expr** and assembles: Norm[v, Infinity] → Max[Abs[v_i]]; otherwise (default p = 2, or "Frobenius" → 2, or a user p) → Power[Sum[Abs[v_i]^p], 1/p]. Every Abs, Power, Plus, Max is created and run through the evaluator (eval_and_free), so exact/symbolic/complex entries produce exact symbolic norms.

Limits. Genuine matrix p-norms (e.g. the spectral 2-norm via the largest singular value) are not implemented — a rank-≥2 argument with a non-"Frobenius" p falls through to NULL and the call stays symbolic. Jagged arrays (rank < 0) also return NULL.

  • Protected.
  • For scalars, Norm[z] is Abs[z].
  • For vectors, Norm[v] defaults to the 2-norm: Sqrt[v . Conjugate[v]].
  • For vectors, Norm[v, p] is Total[Abs[v]^p]^(1/p).
  • For vectors, Norm[v, Infinity] is the $\infty$-norm given by Max[Abs[v]].
  • Norm[m, "Frobenius"] gives the Frobenius norm of a matrix m.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Norm[{3, 4}]
Out[1]= 5

A symbolic 2-norm is kept exact, written with Abs so it is valid for complex components too:

In[1]:= Norm[{a, b, c}]
Out[1]= Sqrt[Abs[a]^2 + Abs[b]^2 + Abs[c]^2]

The same vector under different p-norms — the 1-norm, the max (Infinity) norm, and the exact 3-norm:

In[1]:= Norm[{1, 2, 3, 4}, 1]
Out[1]= 10

In[2]:= Norm[{1, 2, 3, 4}, Infinity]
Out[2]= 4

In[3]:= Norm[{1, 2, 3, 4}, 3]
Out[3]= 10^(2/3)

The norm of a complex vector, taken to 40 significant digits:

In[1]:= N[Norm[{1, 2, 3}], 40]
Out[1]= 3.7416573867739413855837487323165493017559

Notes

Norm[expr] gives the 2-norm of a scalar or vector; Norm[expr, p] gives the p-norm (Abs for scalars; (Sum |xi|^p)^(1/p) for vectors with 1 <= p < Infinity; Max[Abs[expr]] for p == Infinity). Results stay exact for exact input, and symbolic components are written with Abs so the formula is valid for complex entries.