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SimpleMatrixFct
This section describes the usage of the simple simple matrix operations Add/sub Mul/Transpose Functions applied to each element Mean and sum
Most of the functions have two forms. One returns a complete new matrix object the other one operates on the calling object and replaces it's memory by the result. The functions also come in two flavors one with interfaces and one without. Do not mix them in code!
The matrix addition and matrix subtraction is simply a value by
value operation on each element so an addition is defined as
c[x, y] = a[x, y] + b[x, y] and the element wise multiplication as
c[x, y] = a[x, y] * b[x, y]
procedure AddInplace(Value : TDoubleMatrix); overload;
function Add(Value : TDoubleMatrix) : TDoubleMatrix; overload;
procedure SubInPlace(Value : TDoubleMatrix); overload;
function Sub(Value : TDoubleMatrix) : TDoubleMatrix; overload;
procedure AddInplace(Value : IMatrix); overload;
function Add(Value : IMatrix) : TDoubleMatrix; overload;
procedure SubInPlace(Value : IMatrix); overload;
function Sub(Value : IMatrix) : TDoubleMatrix; overload;
procedure ElementWiseMultInPlace(Value : TDoubleMatrix); overload;
function ElementWiseMult(Value : TDoubleMatrix) : TDoubleMatrix; overload;
procedure ElementWiseMultInPlace(Value : IMatrix); overload;
function ElementWiseMult(Value : IMatrix) : TDoubleMatrix; overload;
The base matrix multiplication algorithm is implemented as a recursive version of the Strassen multiplication until a certain matrix size is reached. Then the algorithm changes from the general matrix multiplication algorithm to a blocked version which internally still uses the Strassen algorithm. Use the global variable BlockedMatrixMultSize to change the default settings.
Note: One can switch the libs default behaviour by using the InitMathFunctions in OptimizedFuncs.pas to setup the multiplication mode as well as turning SSE optimizations completely off.
Despite the standard multiplcation there also exists a number of often used multiplication derivaties that implicitly transpose one of the matrices.
One element is calculated as c[i, j] = sum_n(a[n,j]*b[i,n])
// multT1: dest = mt1' * mt2 mt1' = mt1.transpose
procedure MultInPlaceT1(Value : TDoubleMatrix); overload;
function MultT1(Value : TDoubleMatrix) : TDoubleMatrix; overload;
// multT2: dest = mt1 * mt2' mt2 = mt2.transpose
procedure MultInPlaceT2(Value : TDoubleMatrix); overload;
function MultT2(Value : TDoubleMatrix) : TDoubleMatrix; overload;
procedure AddInplace(Value : IMatrix); overload;
function Add(Value : IMatrix) : TDoubleMatrix; overload;
procedure SubInPlace(Value : IMatrix); overload;
function Sub(Value : IMatrix) : TDoubleMatrix; overload;
procedure MultInPlace(Value : IMatrix); overload;
function Mult(Value : IMatrix) : TDoubleMatrix; overload;
procedure MultInPlaceT1(Value : IMatrix); overload;
function MultT1(Value : IMatrix) : TDoubleMatrix; overload;
procedure MultInPlaceT2(Value : IMatrix); overload;
function MultT2(Value : IMatrix) : TDoubleMatrix; overload;
procedure TransposeInPlace;
function Transpose : TDoubleMatrix;
Element wise functions are simply functions applied to each element of the matrix.
procedure AddAndScaleInPlace(const Offset, Scale : double);
function AddAndScale(const Offset, Scale : double) : TDoubleMatrix;
function Scale(const Value : double) : TDoubleMatrix;
procedure ScaleInPlace(const Value : Double);
function ScaleAndAdd(const Offset, Scale : double) : TDoubleMatrix;
procedure ScaleAndAddInPlace(const Offset, Scale : double);
function SQRT : TDoubleMatrix;
procedure SQRTInPlace;
procedure ElementwiseFuncInPlace(func : TMatrixFunc); overload;
function ElementwiseFunc(func : TMatrixFunc) : TDoubleMatrix; overload;
procedure ElementwiseFuncInPlace(func : TMatrixObjFunc); overload;
function ElementwiseFunc(func : TMatrixObjFunc) : TDoubleMatrix; overload;
procedure ElementwiseFuncInPlace(func : TMatrixMtxRefFunc); overload;
function ElementwiseFunc(func : TMatrixMtxRefFunc) : TDoubleMatrix; overload;
procedure ElementwiseFuncInPlace(func : TMatrixMtxRefObjFunc); overload;
function ElementwiseFunc(func : TMatrixMtxRefObjFunc) : TDoubleMatrix; overload;
procedure MaskedSetValue(const Mask : Array of boolean; const newVal : double);
Notes:
- The difference of AddAndScale and ScaleAndAdd is the order of execution. The AddAndScale routine first adds a value and then scales it (and vice versa)
- The routine SQRT applies the root function on each element separately don't confuse it with the root of a matrix.
- The MaskedSetValue treats the matrix as a vector! So the first row of the matrix is reflected in the first n elements of the mask.
Multiplying two matrices
procedure TestIMatrix.TestMult2;
const x : Array[0..8] of double = (1, 2, 3, 4, 5, 6, 7, 8, 9);
y : Array[0..8] of double = (0, 1, 2, 3, 4, 5, 6, 7, 8);
res : Array[0..8] of double = (24, 30, 36, 51, 66, 81, 78, 102, 126);
var m1, m2 : IMatrix;
m3 : IMatrix;
begin
m1 := TDoubleMatrix.Create;
m1.Assign(x, 3, 3);
m2 := TDoubleMatrix.Create;
m2.Assign(y, 3, 3);
m3 := m1.Mult(m2);
Check(CheckMtx(res, m3.SubMatrix, 3, 3), 'Error multiplication failed');
end;
Transposition/multiplication of a matrix
procedure TransposeMult;
var mtx : IMatrix;
m1, m2 : IMatrix;
const x : Array[0..8] of double = (1, 2, 3, 4, 5, 6, 7, 8, 9);
begin
m1 := TDoubleMatrix.Create;
m1.Assign(x, 3, 3);
m2 := m1.Transpose;
mtx := m2.Mult(m1);
// one can also use the transposition mult function
m2.Assign(m1);
mtx := m2.MultT1(m1);
end;
Applying a function to each element of the matrix
procedure sinxDivX(var x : double);
begin
if SameValue(x, 0, 1e-7)
then
x := 1
else
x := sin(x)/x;
end;
procedure ApplyFunc;
var m1 : IMatrix;
m2 : IMatrix;
i : integer;
begin
m1 := TDoubleMatrix.Create(100, 1);
for i := 0 to 99 do
m1[i, 0] := -pi + pi/100;
m2 := m1.ElementwiseFunc(sinxDivX);
end;