Algebraic Operations

Hereinafter the following convention is used:

– small Greek letters are used to denote scalars;
– small Latin letters are used to denote vectors;
– capital Latin letters are used to denote matrices.

This topic contains the following sections:

Unary operations
Addition
Subtraction
Multiplication
Division
Overloaded operators
See Also

Unary operations

This section describes methods implementing unary operations, i.e. operations performed with one vector.

Operation	Description	Performance
plus	Creates a copy of the original vector In place: PlusInPlace Returning result: Plus Out of place: Plus(Vector)
minus	Negates this vector. If the vector is , then the result of this operation will be In place: MinusInPlace Returning result: Minus Out of place: Minus(Vector)

Operation

Description

Performance

plus

Creates a copy of the original vector

minus

Negates this vector. If the vector is VVector , then the result of this operation will be VMinus

Addition

Addition operation allows adding a vector or a scalar to the given vector.

Operation	Description	Performance
vector-scalar addition	Calculates the sum of a scalar and a vector (or a vector and a scalar – as you know the addition is commutative operation). This operation is defined as: given: then: In place: AddInPlace(Double) Returning result: Add(Double) Out of place: Add(Double, Vector)
vector-vector addition	Adds the given vector to this vector. This operation is defined as: given: then: In place: AddInPlace(Vector) Returning result: Add(Vector) Out of place: Add(Vector, Vector)

Operation

Description

Performance

vector-scalar addition

Calculates the sum of a scalar and a vector (or a vector and a scalar – as you know the addition is commutative operation). This operation is defined as:

given:

then:

In place:

method AddInPlace(Double)

Returning result:

method Add(Double)

Out of place:

method Add(Double, Vector)

vector-vector addition

Adds the given vector to this vector. This operation is defined as:

given:

then:

In place:

method AddInPlace(Vector)

Returning result:

method Add(Vector)

Out of place:

method Add(Vector, Vector)

Subtraction

Subtraction allows calculating the difference between a vector and a scalar or between two vectors.

Operation	Description	Performance
vector-scalar subtraction	The difference of a vector and a scalar: In place: SubtractInPlace(Double) Returning result: Subtract(Double) Out of place: Subtract(Double, Vector) The difference of a scalar and a vector: In place: LeftSubtractInPlace(Double) Returning result: LeftSubtract(Double) Out of place: LeftSubtract(Double, Vector)
vector-vector subtraction	The difference of the given vector and another vector: then: In place: SubtractInPlace(Vector) Returning result: Subtract(Vector) Out of place: Subtract(Vector, Vector) The opposite operation: In place: LeftSubtractInPlace(Vector) Returning result: LeftSubtract(Vector) Out of place: LeftSubtract(Vector, Vector)

Operation

Description

Performance

vector-scalar subtraction

The difference of a vector and a scalar:

In place:

method SubtractInPlace(Double)

Returning result:

method Subtract(Double)

Out of place:

method Subtract(Double, Vector)

The difference of a scalar and a vector:

In place:

method LeftSubtractInPlace(Double)

Returning result:

method LeftSubtract(Double)

Out of place:

method LeftSubtract(Double, Vector)

vector-vector subtraction

The difference of the given vector and another vector:

then:

In place:

method SubtractInPlace(Vector)

Returning result:

method Subtract(Vector)

Out of place:

method Subtract(Vector, Vector)

The opposite operation:

In place:

method LeftSubtractInPlace(Vector)

Returning result:

method LeftSubtract(Vector)

Out of place:

method LeftSubtract(Vector, Vector)

Multiplication

Multiplication of a vector and a scalar.

Operation	Description	Performance
vector-scalar multiplication	Multiplies this vector by the given scalar: In place: MultiplyInPlace(Double) Returning result: Multiply(Double) Out of place: Multiply(Double, Vector)

Operation

Description

Performance

vector-scalar multiplication

Multiplies this vector by the given scalar:

In place:

method MultiplyInPlace(Double)

Returning result:

method Multiply(Double)

Out of place:

method Multiply(Double, Vector)

Division

Division of a vector by a scalar.

Operation	Description	Performance
vector-scalar division	Divides this vector by the given scalar: In place: DivideInPlace(Double) Returning result: Divide(Double) Out of place: Divide(Double, Vector)

Operation

Description

Performance

vector-scalar division

Divides this vector by the given scalar:

In place:

method DivideInPlace(Double)

Returning result:

method Divide(Double)

Out of place:

method Divide(Double, Vector)

Overloaded operators

The described algebraic operations methods can be replaced by mathematical signs of this operations: +, -, *, / like in the example below:

Copy

1Vector u = new Vector(4);
2Vector v = new Vector(4);
3
4//vector sum
5Vector sum = u + v;
6//vector-scalar sum
7
8double a = 2;
9sum = u + a + v;

Other Resources

Vector Class

Pointwise Operations

Copying And Conversion

Vector Products

Vector Norms

Extensions

Floating Point Arithmetic

Input And Output