Function Repository Resource:

PowerTotal

Source Notebook

Compute the total of a list of numbers all taken to some power

Contributed by: Seth J. Chandler

ResourceFunction["PowerTotal"][y,n,x]

takes x to the y^th power and then applies the Total function using level specification n.

ResourceFunction["PowerTotal"][y,n]

represents an operator form of ResourceFunction["PowerTotal"] that can be applied to an expression x.

ResourceFunction["PowerTotal"][y]

represents an operator form of ResourceFunction["PowerTotal"] that, when applied to x, computes the total at the top level.

ResourceFunction["PowerTotal"][]

represents an operator form of ResourceFunction["PowerTotal"] that, when applied to x, takes the sum of the squares.

Details and Options

Both x and y can be real, complex or symbolic.

The level specification n may take the form of:

an integer n, meaning the summation is performed at all levels down to level n;

an integer n wrapped with List, meaning the summation is performed at level n only; and

a List of two integers {n₁,n₂}, meaning the summation is performed at all levels n₁ through n₂.

See the documentation for Total for a more complete explanation of how the level specification works there.

Examples

Basic Examples (4)

Compute the sum of the squares of a list:

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Compute the sum of the cubes of a list with symbolic parts:

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Compute the sum of the cubes of a symbolic array:

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Create an operator which when confronted with an expression computes the sum of its square roots:

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Scope (5)

The power may be complex, as may the list:

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The level specification can affect the results when the data has more than one dimension:

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The default behavior is to apply at level 1:

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Apply the total down to level 2:

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Apply the total in the last two dimensions:

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Applications (3)

Use PowerTotal to conduct ordinary least squares linear regression by finding the values of two parameters a and b that minimize the sum of the squared distances between the actual value of the independent variable and a value that depends on a and b:

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Use PowerTotal to perform "Tikhonov" (ridge) regression:

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Use PowerTotal to perform "LASSO" regression:

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Properties and Relations (2)

PowerTotal[] is the same as the square of the results from the Norm function if the arguments it confronts are real-valued, but is not necessarily the same if the values it confronts are complex:

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$ResourceFunction["PowerTotal"][][{3, 4}] == Norm[{3, 4}]^2$