From 2f6bd3c3b166de8beb66a025a8b5f6a7b23503e0 Mon Sep 17 00:00:00 2001 From: Olivier Couet <olivier.couet@cern.ch> Date: Wed, 22 May 2013 11:07:22 +0200 Subject: [PATCH] All the ugly pictures used to render formulas are replaced with nice Latex --- docbook/users-guide/LinearAlgebra.md | 2 - docbook/users-guide/MathLibraries.md | 991 ++++++++++++---------- docbook/users-guide/pictures/03000128.png | Bin 12453 -> 11593 bytes docbook/users-guide/pictures/030001A1.png | Bin 74528 -> 366448 bytes 4 files changed, 561 insertions(+), 432 deletions(-) diff --git a/docbook/users-guide/LinearAlgebra.md b/docbook/users-guide/LinearAlgebra.md index e7a401b162a..f80958d780c 100644 --- a/docbook/users-guide/LinearAlgebra.md +++ b/docbook/users-guide/LinearAlgebra.md @@ -1268,5 +1268,3 @@ calls to constructors and destructors in the `C++` packages versus a `C` library like `GSL`.  - - diff --git a/docbook/users-guide/MathLibraries.md b/docbook/users-guide/MathLibraries.md index 712717c1c3f..8ede5d8b79c 100644 --- a/docbook/users-guide/MathLibraries.md +++ b/docbook/users-guide/MathLibraries.md @@ -54,8 +54,8 @@ Pseudo-random numbers are generated using a linear congruential random generator. The multipliers used are the same of the BSD `rand()` random generator. Its sequence is: - with `=1103515245`, -= 12345 and =2^31^. +$x_{n+1} = (ax_n + c) \bmod{m}$ with $a =1103515245$, +$c = 12345$ and $m =2^{31}$. This type of generator uses a state of only a 32 bit integer and it has a very short period, 2^31^,about 10^9^, which can be exhausted in just @@ -134,18 +134,19 @@ The method `Rndm()` is used for generating a pseudo-random number distributed between 0 and 1 as shown in the following example: ``` {.cpp} -// use default seed (same random numbers will be generated each time) -TRandom3 r; // generate a number in interval ]0,1] (0 is excluded) -r.Rndm(); -double x[100]; -r.RndmArray(100,x); // generate an array of random numbers in ]0,1] -TRandom3 rdm(111); // construct with a user-defined seed -// use 0: a unique seed will be automatically generated using TUUID -TRandom1 r1(0); -TRandom2 r2(0); -TRandom3 r3(0); -// use a seed generated using machine clock (different every second) -TRandom r0(0); + // use default seed + // (same random numbers will be generated each time) + TRandom3 r; // generate a number in interval ]0,1] (0 is excluded) + r.Rndm(); + double x[100]; + r.RndmArray(100,x); // generate an array of random numbers in ]0,1] + TRandom3 rdm(111); // construct with a user-defined seed + // use 0: a unique seed will be automatically generated using TUUID + TRandom1 r1(0); + TRandom2 r2(0); + TRandom3 r3(0); + // seed generated using machine clock (different every second) + TRandom r0(0); ``` ### Random Number Distributions @@ -161,9 +162,10 @@ require several random numbers. ``` {.cpp} TRandom3 r; -// generate a gaussian distributed number with mu=0, sigma=1 (default values) -double x1 = r.Gaus(); -double x2 = r.Gaus(10,3); // use mu = 10, sigma = 3; + // generate a gaussian distributed number with: + // mu=0, sigma=1 (default values) + double x1 = r.Gaus(); + double x2 = r.Gaus(10,3); // use mu = 10, sigma = 3; ``` The following table shows the various distributions that can be @@ -247,13 +249,14 @@ can be used as following: <http://statistik.wu-wien.ac.at/unuran/doc/unuran.html> ): ``` {.cpp} -TUnuran unr; -//initialize unuran to generate normal random numbers using a "arou" method -unr.Init("normal()","method=arou"); -... -// sample distributions N times (generate N random numbers) -for (int i = 0; i<N; ++i) -double x = unr.Sample(); + TUnuran unr; + // initialize unuran to generate normal random numbers using + // a "arou" method + unr.Init("normal()","method=arou"); + ... + // sample distributions N times (generate N random numbers) + for (int i = 0; i<N; ++i) + double x = unr.Sample(); ``` - For continuous 1D distribution object via the class @@ -264,13 +267,15 @@ double x = unr.Sample(); generating numbers in a restricted region. ``` {.cpp} -//1D case: create a distribution from two TF1 object pointers pdfFunc -TUnuranContDist dist( pdfFunc); -//initialize unuran passing the distribution and a string defining the method -unr.Init(dist, "method=hinv"); -// sample distribution N times (generate N random numbers) -for (int i = 0; i < N; ++i) -double x = unr.Sample(); + // 1D case: create a distribution from two TF1 object + // pointers pdfFunc + TUnuranContDist dist( pdfFunc); + // initialize unuran passing the distribution and a string + // defining the method + unr.Init(dist, "method=hinv"); + // sample distribution N times (generate N random numbers) + for (int i = 0; i < N; ++i) + double x = unr.Sample(); ``` - For multi-dimensional distribution via the class @@ -278,14 +283,14 @@ double x = unr.Sample(); multi-dimensional pdf. ``` {.cpp} -// Multi- dimensional case from a TF1 (TF2 or TF3) objects -TUnuranMultiContDist dist( pdfFuncMulti); -// the recommended method for multi-dimensional function is "hitro" -unr.Init(dist,"method=hitro"); -// sample distribution N times (generate N random numbers) -double x[NDIM]; -for (int i = 0; i<N; ++i) -unr.SampleMulti(x); + // Multi- dimensional case from a TF1 (TF2 or TF3) objects + TUnuranMultiContDist dist( pdfFuncMulti); + // the recommended method for multi-dimensional function is "hitro" + unr.Init(dist,"method=hitro"); + // sample distribution N times (generate N random numbers) + double x[NDIM]; + for (int i = 0; i<N; ++i) + unr.SampleMulti(x); ``` - For discrete distribution via the class **`TUnuranDiscrDist`**, @@ -293,14 +298,14 @@ unr.SampleMulti(x); probabilities. ``` {.cpp} -// Create distribution from a vector of probabilities -double pv[NSize] = {0.1,0.2,...}; -TUnuranDiscrDist dist(pv,pv+NSize); -// the recommended method for discrete distribution is -unr.Init(dist, "method=dgt"); -// sample N times (generate N random numbers) -for (int i = 0; i < N; ++i) -int k = unr.SampleDiscr(); + // Create distribution from a vector of probabilities + double pv[NSize] = {0.1,0.2,...}; + TUnuranDiscrDist dist(pv,pv+NSize); + // the recommended method for discrete distribution is + unr.Init(dist, "method=dgt"); + // sample N times (generate N random numbers) + for (int i = 0; i < N; ++i) + int k = unr.SampleDiscr(); ``` - For empirical distribution via the class **`TUnuranEmpDist`**. In @@ -312,13 +317,13 @@ int k = unr.SampleDiscr(); un-binned data). ``` {.cpp} -// Create distribution from a set of data -// vdata is an std::vector containing the data -TUnuranEmpDist dist(vdata.begin(),vdata.end()); -unr.Init(dist); -// sample N times (generate N random numbers) -for (int i = 0; i<N; ++i) -double x = unr.Sample(); + // Create distribution from a set of data + // vdata is an std::vector containing the data + TUnuranEmpDist dist(vdata.begin(),vdata.end()); + unr.Init(dist); + // sample N times (generate N random numbers) + for (int i = 0; i<N; ++i) + double x = unr.Sample(); ``` - For some predefined distributions, like `Poisson` and `Binomial`, @@ -327,14 +332,14 @@ double x = unr.Sample(); directly the string interface. ``` {.cpp} -TUnuran unr; -// Initialize unuran to generate normal random numbers from the Poisson -// distribution with parameter mu -unr.InitPoisson(mu); -... -// Sample distributions N times (generate N random numbers) -for (int i = 0; i<N; ++i) -int k = unr.SampleDiscr(); + TUnuran unr; + // Initialize unuran to generate normal random numbers from the + // Poisson distribution with parameter mu + unr.InitPoisson(mu); + ... + // Sample distributions N times (generate N random numbers) + for (int i = 0; i<N; ++i) + int k = unr.SampleDiscr(); ``` Functionality is also provided via the C++ classes for using a different @@ -346,7 +351,7 @@ to UNURAN). Here are the CPU times obtained using the four random classes on an -lxplus machine with an Intel 64 bit architecture and compiled using gcc +`lxplus` machine with an Intel 64 bit architecture and compiled using gcc 3.4: +--------------------+---------------+---------------+---------------+---------------+ @@ -691,10 +696,10 @@ system type. It can be even any vector of a different package, like the CLHEP **`HepThreeVector`** that implements the required signature. ``` {.cpp} -XYZVector v1(1,2,3); -RhoEtaPhiVector r2(v1); -CLHEP::HepThreeVector q(1,2,3); -XYZVector v3(q); + XYZVector v1(1,2,3); + RhoEtaPhiVector r2(v1); + CLHEP::HepThreeVector q(1,2,3); + XYZVector v3(q); ``` #### Coordinate Accessors @@ -703,24 +708,26 @@ All coordinate accessors are available through the class `ROOT::Math::`**`DisplacementVector3D`**: ``` {.cpp} -//returns cartesian components for the cartesian vector v1 -v1.X(); v1.Y(); v1.Z(); -//returns cylindrical components for the cartesian vector v1 -v1.Rho(); v1.Eta(); v1.Phi(); -//returns cartesian components for the cylindrical vector r2 -r2.X(); r2.Y(); r2.Z() + //returns cartesian components for the cartesian vector v1 + v1.X(); v1.Y(); v1.Z(); + //returns cylindrical components for the cartesian vector v1 + v1.Rho(); v1.Eta(); v1.Phi(); + //returns cartesian components for the cylindrical vector r2 + r2.X(); r2.Y(); r2.Z() ``` In addition, all the 3 coordinates of the vector can be retrieved with the `GetCoordinates` method: ``` {.cpp} -double d[3]; -v1.GetCoordinates(d); //fill d array with (x,y,z) components of v1 -r2.GetCoordinates(d); //fill d array with (r,eta,phi) components of r2 -std::vector vc(3); -//fill std::vector with (x,y,z) components of v1 -v1.GetCoordinates(vc.begin(),vc.end()); + double d[3]; + //fill d array with (x,y,z) components of v1 + v1.GetCoordinates(d); + //fill d array with (r,eta,phi) components of r2 + r2.GetCoordinates(d); + std::vector vc(3); + //fill std::vector with (x,y,z) components of v1 + v1.GetCoordinates(vc.begin(),vc.end()); ``` See the reference documentation of @@ -732,9 +739,9 @@ coordinate accessors. One can set only all the three coordinates via: ``` {.cpp} -v1.SetCoordinates(c1,c2,c3); //sets the (x,y,z) for a XYZVector -r2.SetCoordinates(c1,c2,c3); //sets r,theta,phi for a Polar3DVector -r2.SetXYZ(x,y,z); //sets the 3 cartesian components for Polar3DVector + v1.SetCoordinates(c1,c2,c3); // (x,y,z) for a XYZVector + r2.SetCoordinates(c1,c2,c3); // r,theta,phi for a Polar3DVector + r2.SetXYZ(x,y,z); // 3 cartesian components for Polar3DVector ``` Single coordinate setter methods are available for the basic vector @@ -742,26 +749,29 @@ coordinates, like `SetX()` for a **`XYZVector`** or `SetR()` for a polar vector. Attempting to do a `SetX()` on a polar vector will not compile. ``` {.cpp} -XYZVector v1; v1.SetX(1); //OK setting x for a Cartesian vector -Polar3DVector v2; v2.SetX(1); //ERROR: cannot set X for a Polar vector. -//Method will not compile -v2.SetR(1); //OK setting r for a Polar vector + XYZVector v1; + v1.SetX(1); //OK setting x for a Cartesian vector + Polar3DVector v2; + v2.SetX(1); //ERROR: cannot set X for a Polar vector. + //Method will not compile + v2.SetR(1); //OK setting r for a Polar vector ``` In addition, there are setter methods from C arrays or iterator ``` {.cpp} -double d[3] = {1.,2.,3.}; -XYZVector v; -v.SetCoordinates(d); //set (x,y,z) components of v using values from d + double d[3] = {1.,2.,3.}; + XYZVector v; + // set (x,y,z) components of v using values from d + v.SetCoordinates(d); ``` or, for example, from an `std::vector` using the iterator ``` {.cpp} -std::vector w(3); -// set (x,y,z) components of v using values from w -v.SetCoordinates(w.begin(),w.end()); + std::vector w(3); + // set (x,y,z) components of v using values from w + v.SetCoordinates(w.begin(),w.end()); ``` #### Arithmetic Operations @@ -772,16 +782,16 @@ different coordinate system types: (`v1,v2` are any type of of `v1`; `a` is a scalar value) ``` {.cpp} -v1 += v2; -v1 -= v2; -v1 = - v2; -v1 *= a; -v1 /= a; -v2 = a * v1; -v2 = v1 / a; -v2 = v1 * a; -v3 = v1 + v2; -v3 = v1 - v2; + v1 += v2; + v1 -= v2; + v1 = - v2; + v1 *= a; + v1 /= a; + v2 = a * v1; + v2 = v1 / a; + v2 = v1 * a; + v3 = v1 + v2; + v3 = v1 - v2; ``` #### Comparison @@ -790,8 +800,8 @@ For `v1` and `v2` of the same type (same coordinate system and same scalar type): ``` {.cpp} -v1 == v2; -v1 != v2; + v1 == v2; + v1 != v2; ``` #### Dot and Cross Product @@ -800,9 +810,9 @@ We support the dot and cross products, through the `Dot()` and `Cross()` method, with any vector (`q`) implementing `x()`, `y()` and `z()`. ``` {.cpp} -XYZVector v1(x,y,z); -double s = v1.Dot(q); -XYZVector v2 = v1.Cross(q); + XYZVector v1(x,y,z); + double s = v1.Dot(q); + XYZVector v2 = v1.Cross(q); ``` Note that the multiplication between two vectors using the operator `*` @@ -811,7 +821,7 @@ is not supported because it is ambiguous. #### Other Methods ``` {.cpp} -XYZVector u = v1.Unit(); //return unit vector parallel to v1 + XYZVector u = v1.Unit(); //return unit vector parallel to v1 ``` ### Example: 3D Point Classes @@ -853,8 +863,8 @@ template class **`ROOT::Math`**`::`**`PositionVector3D`**: The following declarations are available: ``` {.cpp} -XYZPoint p1; //an empty vector (x=0, y=0, z=0) -XYZPoint p2(1,2,3); // + XYZPoint p1; //an empty vector (x=0, y=0, z=0) + XYZPoint p2(1,2,3); // ``` Note that each point type is constructed by passing its coordinate @@ -866,10 +876,10 @@ or even any vector of a different package, like the CLHEP **`HepThreePoint`** that implements the required signatures. ``` {.cpp} -XYZPoint p1(1,2,3); -RhoEtaPHiPoint r2(v1); -CLHEP::HepThreePoint q(1,2,3); -XYZPoint p3(q); + XYZPoint p1(1,2,3); + RhoEtaPHiPoint r2(v1); + CLHEP::HepThreePoint q(1,2,3); + XYZPoint p3(q); ``` #### Coordinate Accessors and Setter Methods @@ -886,14 +896,15 @@ same type; `v1` and `v2` are `ROOT::Math::`**`DisplacementVector3D`** objects). ``` {.cpp} -p1 += v1; -p1 -= v1; -p3 = p1 + v1; //p1 and p3 are the same type -p3 = v1 + p1; //p3 is based on the same coordinate system as v1 -p3 = p1 - v1; -p3 = v1 - p1; -v2 = p1 - p2; //difference between points returns a vector v2 based on the -//same coordinate system as p1 + p1 += v1; + p1 -= v1; + p3 = p1 + v1; // p1 and p3 are the same type + p3 = v1 + p1; // p3 is based on the same coordinate system as v1 + p3 = p1 - v1; + p3 = v1 - p1; + v2 = p1 - p2; // difference between points returns a + // vector v2 based on the + //same coordinate system as p1 ``` Note that the addition between two points is **NOT** possible and the @@ -940,9 +951,12 @@ The metric used for all the LorentzVector is (`-,-,-,+`) . The following declarations are available: ``` {.cpp} -XYZTVector v1; //create an empty vector (x=0, y=0, z=0, t=0) -XYZTVector v2(1,2,3,4); //vector with x=1, y=2, z=3, t=4 -PtEtaPhiEVector v3(1,2,PI,5); //vector with pt=1, eta=2, phi=PI, E=5 + // create an empty vector (x=0, y=0, z=0, t=0) + XYZTVector v1; + // vector with x=1, y=2, z=3, t=4 + XYZTVector v2(1,2,3,4); + // vector with pt=1, eta=2, phi=PI, E=5 + PtEtaPhiEVector v3(1,2,PI,5); ``` Note that each type of vector is constructed by passing its coordinate @@ -956,10 +970,10 @@ different coordinate system or any vector of a different package, like the CLHEP **`HepLorentzVector`** that implements the required signature. ``` {.cpp} -XYZTVector v1(1,2,3,4); -PtEtaPhiEVector v2(v1); -CLHEP::HepLorentzVector q(1,2,3,4); -XYZTVector v3(q); + XYZTVector v1(1,2,3,4); + PtEtaPhiEVector v2(v1); + CLHEP::HepLorentzVector q(1,2,3,4); + XYZTVector v3(q); ``` #### Coordinate Accessors @@ -968,24 +982,27 @@ All the same coordinate accessors are available through the interface of `ROOT::Math::`**`LorentzVector`**. For example: ``` {.cpp} -//returns cartesian components for the cartesian vector v1 -v1.X(); v1.X(); v1.Z(); v1.T(); -//returns cartesian components for the cylindrical vector v2 -v2.Px(); v2.Py(); v2.Pz(); v2.E(); -//returns other components for the cartesian vector v1 -v1.Pt(); v1.Eta(); v1.Phi(); v1.M() + //returns cartesian components for the cartesian vector v1 + v1.X(); v1.X(); v1.Z(); v1.T(); + //returns cartesian components for the cylindrical vector v2 + v2.Px(); v2.Py(); v2.Pz(); v2.E(); + //returns other components for the cartesian vector v1 + v1.Pt(); v1.Eta(); v1.Phi(); v1.M() ``` In addition, all 4 vector coordinates can be retrieved with the `GetCoordinates` method: ``` {.cpp} -double d[4]; -v1.GetCoordinates(d); //fill d array with (x,y,z,t) components of v1 -v2.GetCoordinates(d); //fill d array with (pt,eta,phi,e) components of v2 -std::vector w(4); -v1.GetCoordinates(w.begin(),w.end()); //fill std::vector with (x,y,z,t) -//components of v1 + double d[4]; + //fill d array with (x,y,z,t) components of v1 + v1.GetCoordinates(d); + //fill d array with (pt,eta,phi,e) components of v2 + v2.GetCoordinates(d); + std::vector w(4); + //fill std::vector with (x,y,z,t) + v1.GetCoordinates(w.begin(),w.end()); + //components of v1 ``` To get information on all the coordinate accessors see the @@ -996,9 +1013,12 @@ To get information on all the coordinate accessors see the One can set only all the three coordinates via: ``` {.cpp} -v1.SetCoordinates(c1,c2,c3,c4); //sets the (x,y,z,t) for a XYZTVector -v2.SetCoordinates(c1,c2,c3,c4); //sets pt,eta,phi,e for a PtEtaPhiEVector -v2.SetXYZ(x,y,z,t); //sets cartesian components for PtEtaPhiEVector + //sets the (x,y,z,t) for a XYZTVector + v1.SetCoordinates(c1,c2,c3,c4); + //sets pt,eta,phi,e for a PtEtaPhiEVector + v2.SetCoordinates(c1,c2,c3,c4); + //sets cartesian components for PtEtaPhiEVector + v2.SetXYZ(x,y,z,t); ``` Single coordinate setter methods are available for the basic vector @@ -1007,26 +1027,29 @@ coordinates, like `SetX()` for a `XYZTVector` or `SetPt()` for a vector will not compile. ``` {.cpp} -XYZTVector v1; v1.SetX(1); //OK setting x for a cartesian vector -PtEtaPhiEVector v2; v2.SetX(1); //ERROR: cannot set X for a non-cartesian -//vector. Method will not compile. -v2.SetR(1) // OK setting Pt for a PtEtaPhiEVector vector + XYZTVector v1; + v1.SetX(1); //OK setting x for a cartesian vector + PtEtaPhiEVector v2; + v2.SetX(1); //ERROR: cannot set X for a non-cartesian + //vector. Method will not compile. + v2.SetR(1) // OK setting Pt for a PtEtaPhiEVector vector ``` In addition, there are setter methods from C arrays or iterators. ``` {.cpp} -double d[4] = {1.,2.,3.,4.}; -XYZTVector v; -v.SetCoordinates(d); //set (x,y,z,t) components of v using values from d + double d[4] = {1.,2.,3.,4.}; + XYZTVector v; + //set (x,y,z,t) components of v using values from d + v.SetCoordinates(d); ``` or for example from an `std::vector `using the iterators ``` {.cpp} -std::vector w(4); -//set (x,y,z,t) components of v using values from w -v.SetCoordinates(w.begin(),w.end()); + std::vector w(4); + //set (x,y,z,t) components of v using values from w + v.SetCoordinates(w.begin(),w.end()); ``` #### Arithmetic Operations @@ -1038,34 +1061,35 @@ vector of the same type, `q `is a generic Lorentz vector implementing float, int, etc.) . ``` {.cpp} -v += q; -v -= q; -v = -q; -v *= a; -v /= a; -w = v + q; -w = v - q; -w = v * a; -w = a * v; -w = v / a; + v += q; + v -= q; + v = -q; + v *= a; + v /= a; + w = v + q; + w = v - q; + w = v * a; + w = a * v; + w = v / a; ``` #### Comparison ``` {.cpp} -v == w; -v != w; + v == w; + v != w; ``` #### Other Methods ``` {.cpp} -a = v.Dot(q); //dot product in metric(+,+,+,-) of 2 LorentzVectors -XYZVector s = v.Vect() //return the spatial components (x,y,z) -v.Beta(); //return beta and gamma value (vector must -v.Gamma() // be time-like otherwise result is meaningless) -XYZVector b = v.BoostToCM(); //return boost vector which will bring the Vector -//in its mas frame (P=0) + a = v.Dot(q); //dot product in metric(+,+,+,-) of 2 LorentzVectors + XYZVector s = v.Vect() //return the spatial components (x,y,z) + v.Beta(); //return beta and gamma value (vector must + v.Gamma() // be time-like otherwise result is meaningless) + XYZVector b = v.BoostToCM(); //return boost vector which will bring + //the Vector + //in its mas frame (P=0) ``` ### Example: Vector Transformations @@ -1074,9 +1098,9 @@ XYZVector b = v.BoostToCM(); //return boost vector which will bring the Vector Transformation classes are grouped in rotations (in three dimensions), Lorentz transformations and Poincarre transformations, which are translation`/`rotation combinations. Each group has several members -which may model physically equivalent transformations but with different -internal representations. All the classes are non-template and use -double precision as the scalar type. The following types of +which may model physically equivalent transformations but with +different internal representations. All the classes are non-template +and use double precision as the scalar type. The following types of transformation classes are defined: 3D rotations: @@ -1134,11 +1158,13 @@ number of scalar arguments matching the number (and order of components). ``` {.cpp} -Rotation3D rI; //a summy rotation (Identity matrix) -RotationX rX(PI); //a RotationX with an angle PI -EulerAngles rE(phi,theta,psi); //an Euler rotation with phi,theta,psi angles -XYZVector u(ux,uy,uz); -AxisAngle rA(u,delta); //a rotation based on direction u, angle delta + Rotation3D rI; //a summy rotation (Identity matrix) + RotationX rX(PI); //a RotationX with an angle PI + EulerAngles rE(phi,theta,psi); //an Euler rotation with phi, + //theta,psi angles + XYZVector u(ux,uy,uz); + AxisAngle rA(u,delta); //a rotation based on direction u, + //angle delta ``` In addition, all rotations and transformations (other than the axial @@ -1146,36 +1172,42 @@ rotations) and transformations are constructible from (`begin`,`end`) iterators or from pointers behave like iterators. ``` {.cpp} -double data[9]; -Rotation3D r(data,data+9); //create a rotation from a rotation matrix -std::vector w(12); -Transform3D t(w.begin(),w.end()); //create Transform3D from std::vector content + double data[9]; + //create a rotation from a rotation matrix + Rotation3D r(data,data+9); + std::vector w(12); + //create Transform3D from std::vector content + Transform3D t(w.begin(),w.end()); ``` All rotations, except the axial rotations, are constructible and assigned from any other type of rotation (including the axial): ``` {.cpp} -//create a rotation 3D from a rotation along X axis of angle PI -Rotation3D r(ROOT::Math::RotationX(PI)); + //create a rotation 3D from a rotation along X axis of angle PI + Rotation3D r(ROOT::Math::RotationX(PI)); -//construct an Euler rotation from A Rotation3D -EulerAngles r2(r); + //construct an Euler rotation from A Rotation3D + EulerAngles r2(r); -//assign an Axis rotation from an Euler Rotation -AxisAngle r3; r3 = r2; + //assign an Axis rotation from an Euler Rotation + AxisAngle r3; r3 = r2; ``` **`Transform3D`** (rotation + translation) can be constructed from a rotation and a translation vector: ``` {.cpp} -Rotation3D r; -XYZVector v; -Transform3D t1(r,v); //construct from rotation and then translation -Transform3D t2(v,r); //construct inverse from first translation then rotation -Transform3D t3(r); //construct from only a rotation (zero translation) -Transform3D t4(v); //construct from only translation (identity rotation) + Rotation3D r; + XYZVector v; + Transform3D t1(r,v); //construct from rotation and then + //translation + Transform3D t2(v,r); //construct inverse from first translation + //then rotation + Transform3D t3(r); //construct from only a rotation + //(zero translation) + Transform3D t4(v); //construct from only translation + //(identity rotation) ``` #### Operations @@ -1184,10 +1216,10 @@ All transformations can be applied to vector and points using the operator \* or using the operator() ``` {.cpp} -XYZVector v1(...); -Rotation3D r(...); -XYZVector v2 = r*v1; //rotate vector v1 using r -v2 = r(v1); //equivalent + XYZVector v1(...); + Rotation3D r(...); + XYZVector v2 = r*v1; //rotate vector v1 using r + v2 = r(v1); //equivalent ``` Transformations can be combined using the operator `*`. Rotation, @@ -1197,9 +1229,10 @@ will be a **`Transform3D`** class. Note that the rotations are not commutative, the order is then important. ``` {.cpp} -Rotation3D r1(...); -Rotation3D r2(...); -Rotation3D r3 = r2*r1; //a combine rotation r3 by applying first r1 then r2 + Rotation3D r1(...); + Rotation3D r2(...); + Rotation3D r3 = r2*r1; //a combine rotation r3 by + //applying first r1 then r2 ``` We can combine rotations of different types, like `Rotation3D` with any @@ -1207,18 +1240,19 @@ other type of rotations. The product of two different axial rotations returns a `Rotation3D`: ``` {.cpp} -RotationX rx(1.); -RotationY ry(2.); -Rotation3D r = ry * rx; //rotation along X and then Y axis + RotationX rx(1.); + RotationY ry(2.); + Rotation3D r = ry * rx; //rotation along X and then Y axis ``` It is also possible to invert all the transformation or return their inverse: ``` {.cpp} -Rotation3D r1(...); -r1.Invert(); //invert the rotation modifying its content -Rotation3D r2 =r1.Inverse(); //return the inverse in a new rotation class + Rotation3D r1(...); + r1.Invert(); //invert the rotation modifying its content + Rotation3D r2 =r1.Inverse(); //return the inverse in a new + //rotation class ``` We have used rotation as examples, but all these operations can be @@ -1231,17 +1265,17 @@ They can be used to retrieve all the scalar values on which the transformation is based. ``` {.cpp} -RotationX rx; -rx.SetComponents(1.); //set agle of the X rotation -double d[9] = {........}; -Rotation3D r; -r.SetComponents(d,d+9); //set 9 components of 3D rotation -double d[16]; -LorentzRotation lr; -lr.GetComponents(d,d+16); //get 16 components of a LorentzRotation -TMatrixD(3,4) m; -Transform3D t; -t.GetComponens(m); //fill 3x4 matrix with components of t + RotationX rx; + rx.SetComponents(1.); //set agle of the X rotation + double d[9] = {........}; + Rotation3D r; + r.SetComponents(d,d+9); //set 9 components of 3D rotation + double d[16]; + LorentzRotation lr; + lr.GetComponents(d,d+16); //get 16 components of a LorentzRotation + TMatrixD(3,4) m; + Transform3D t; + t.GetComponens(m); //fill 3x4 matrix with components of t ``` The` GetComponents` and `SetComponents` methods can be used with a @@ -1263,27 +1297,29 @@ C array (double`*`) behaves like an iterator. It is then assumed that the coordinates, like (`x,y,z`) will be stored contiguously. ``` {.cpp} -TVectorD r2(N); //ROOT Linear Algebra Vector containing many vectors -XYZVector v2; -//construct vector from x=r[INDEX], y=r[INDEX+1], z=r[INDEX+2] -v2.SetCoordinates(&r2[INDEX],&r2[index]+3); + TVectorD r2(N); //ROOT Linear Algebra Vector containing + //many vectors + XYZVector v2; + //construct vector from x=r[INDEX], y=r[INDEX+1], z=r[INDEX+2] + v2.SetCoordinates(&r2[INDEX],&r2[index]+3); ``` To fill a linear algebra vector from a 3D or 4D vector, with `GetCoordinates()` one can get the internal coordinate data. ``` {.cpp} -HepVector c(3); //CLHEP Linear algebra vector -//fill HepVector c with c[0]=x, c[1]=y, c[2]=z -v2.GetCoordinates(&c[0],&c[index]+3) + HepVector c(3); //CLHEP Linear algebra vector + //fill HepVector c with c[0]=x, c[1]=y, c[2]=z + v2.GetCoordinates(&c[0],&c[index]+3) ``` or using **`TVectorD`**: ``` {.cpp} -double *data[3]; -v2.GetCoordinates(data,data+3); -TVectorD r1(3,data); //create a new Linear Algebra vector copying the data + double *data[3]; + v2.GetCoordinates(data,data+3); + TVectorD r1(3,data); //create a new Linear Algebra vector + //copying the data ``` In the case of transformations, constructor and method to set`/`get @@ -1292,8 +1328,8 @@ matrix data are stored, for example in the case of a Lorentz rotation, from (`0,0`) thru (`3,3`) ``` {.cpp} -TMatrixD(4,4) m; -LorentzRotation r(m); //create Lorentz r + TMatrixD(4,4) m; + LorentzRotation r(m); //create Lorentz r ``` #### Connection to Other Vector Classes @@ -1307,10 +1343,11 @@ assigned from any vector which satisfies the following requisites: methods. ``` {.cpp} -CLHEP::Hep3Vector hv; -XYZVector v1(hv); //create 3D vector from CLHEP 3D Vector -HepGeom::Point3D hp; -XYZPoint p1(hp); //create a 3D p + CLHEP::Hep3Vector hv; + XYZVector v1(hv); //create 3D vector from + //CLHEP 3D Vector + HepGeom::Point3D hp; + XYZPoint p1(hp); //create a 3D p ``` ## MathMore Library @@ -1361,7 +1398,11 @@ License. `MathMore` (and its ROOT CINT dictionary) can be built within ROOT whenever a GSL library is found in the system. The GSL library and header file location can be specified in the ROOT configure script, by -doing: `./configure --with-gsl-incdir=... --with-gsl-libdir=...` +doing: + +``` +./configure --with-gsl-incdir=... --with-gsl-libdir=... +``` `MathMore` can be built also a stand-alone library (without requiring ROOT) downloding the tar file from the Web at this link. In this case @@ -1382,99 +1423,156 @@ in the namespace **`ROOT::Math`**. The most used functions are in the functions in `MathMore` are all using the implementation of the GNU Scientific Library (GSL). The naming of the special functions is the same defined in the C++ -[Technical Report on Standard Library extensions](Technical Report on Standard Library extensions). +[Technical Report on Standard Library extensions](Technical Report on +Standard Library extensions). The special functions are defined in the header file `Math/SpecFunc.h`. ### Special Functions in MathCore - `ROOT::Math::beta(double x,double y) - `evaluates the beta function: -  + $$B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}$$ -- `double ``ROOT::Math::erf(double x)` - evaluates the error function +- `double ROOT::Math::erf(double x)` - evaluates the error function encountered in integrating the normal - distribution: + distribution: + $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt$$ -- `double ``ROOT::Math::erfc(double x)` - evaluates the complementary - error function:  +- `double ROOT::Math::erfc(double x)` - evaluates the complementary + error function: + $$erfc(x) = 1 - erf(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt$$ -- `double ``ROOT::Math::tgamma(double x)` - calculates the gamma - function:  +- `double ROOT::Math::tgamma(double x)` - calculates the gamma + function: + $$\Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt$$ ### Special Functions in MathMore -- `double ``ROOT::Math::assoc_legendre(unsigned l,unsigned m,double x) -`computes - the associated Legendre polynomials (with *`m>=0`, `l>=m` and - `|x|<1)`:* +- `double ROOT::Math::assoc_legendre(unsigned l,unsigned m,double x) -`computes + the associated Legendre polynomials (with `m>=0`, `l>=m` and + `|x|<1)`: + $$P_{l}^{m}(x) = (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{l}(x)$$ -- `double ``ROOT::Math::comp_ellint_1(double k)` - calculates the - complete elliptic integral of the first kind (with `0code>:` +- `double ROOT::Math::comp_ellint_1(double k)` - calculates the + complete elliptic integral of the first kind (with $0 \le k^2 \le 1$: + $$ + K(k) = F(k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{\sqrt{1 - k^2 \sin^2{\theta}}} + $$ -- `double ``ROOT::Math::comp_ellint_2(double k)` - calculates the - complete elliptic integral of the second kind (with `0code>): ` +- `double ROOT::Math::comp_ellint_2(double k)` - calculates the + complete elliptic integral of the second kind (with $0 \le k^2 \le 1$): + $$ + E(k) = E(k , \pi / 2) = \int_{0}^{\pi /2} \sqrt{1 - k^2 \sin^2{\theta}} d \theta + $$ -- `double ``ROOT::Math::comp_ellint_3(double n,double k)` - calculates - the complete elliptic integral of the third kind (with `0code>): ` +- `double ROOT::Math::comp_ellint_3(double n,double k)` - calculates + the complete elliptic integral of the third kind (with $0 \le k^2 \le 1$): + $$ + \Pi (n, k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{(1 - n \sin^2{\theta})\sqrt{1 - k^2 \sin^2{\theta}}} + $$ -- `double ``ROOT::Math::conf_hyperg(double a,double b,double z)` - +- `double ROOT::Math::conf_hyperg(double a,double b,double z)` - calculates the confluent hyper-geometric functions of the first - kind:  - -- `double `**`ROOT::Math::`** - `conf_hypergU(double a,double b,double z) - calculates the confluent hyper-geometric functions of the second kind, known also as Kummer function of the second type. It is related to the confluent hyper-geometric function of the first kind:` - --  - -- `double ``ROOT::Math::cyl_bessel_i(double nu,double x)` - calculates + kind: + $$ + _{1}F_{1}(a;b;z) = \frac{\Gamma(b)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(b+n)} \frac{z^n}{n!} + $$ + +- `double ROOT::Math::conf_hypergU(double a,double b,double z)` - + calculates the confluent hyper-geometric + functions of the second kind, known also as Kummer function of the second type. It is + related to the confluent hyper-geometric function of the first kind: + $$ + U(a,b,z) = \frac{ \pi}{ \sin{\pi b } } \left[ \frac{ _{1}F_{1}(a,b,z) } {\Gamma(a-b+1) } - \frac{ z^{1-b} { _{1}F_{1}}(a-b+1,2-b,z)}{\Gamma(a)} \right] + $$ + +- `double ROOT::Math::cyl_bessel_i(double nu,double x)` - calculates the modified Bessel function of the first kind, also called regular - modified (cylindrical) Bessel function:  + modified (cylindrical) Bessel function: + $$ + I_{\nu} (x) = i^{-\nu} J_{\nu}(ix) = \sum_{k=0}^{\infty} \frac{(\frac{1}{2}x)^{\nu + 2k}}{k! \Gamma(\nu + k + 1)} + $$ -- `double ``ROOT::Math::cyl_bessel_j(double nu,double x)` - calculates +- `double ROOT::Math::cyl_bessel_j(double nu,double x)` - calculates the (cylindrical) Bessel function of the first kind, also called - regular (cylindrical) Bessel function:  + regular (cylindrical) Bessel function: + $$ + J_{\nu} (x) = \sum_{k=0}^{\infty} \frac{(-1)^k(\frac{1}{2}x)^{\nu + 2k}}{k! \Gamma(\nu + k + 1)} + $$ -- `double ``ROOT::Math::cyl_bessel_k(double nu,double x)` - calculates +- `double ROOT::Math::cyl_bessel_k(double nu,double x)` - calculates the modified Bessel function of the second kind, also called - irregular modified (cylindrical) Bessel function for `x>0`, `v>0`: -  + irregular modified (cylindrical) Bessel function for $x > 0$, $v > 0$: + $$ + K_{\nu} (x) = \frac{\pi}{2} i^{\nu + 1} (J_{\nu} (ix) + iN(ix)) = \left\{ \begin{array}{cl} \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin{\nu \pi}} & \mbox{for non-integral $\nu$} \\ \frac{\pi}{2} \lim{\mu \to \nu} \frac{I_{-\mu}(x) - I_{\mu}(x)}{\sin{\mu \pi}} & \mbox{for integral $\nu$} \end{array} \right. + $$ -- `double ``ROOT::Math::cyl_neumann(double nu,double x)` - calculates +- `double ROOT::Math::cyl_neumann(double nu,double x)` - calculates the (cylindrical) Bessel function of the second kind, also called irregular (cylindrical) Bessel function or (cylindrical) Neumann - function:  - -- `double ``ROOT::Math::ellint_1(double k,double phi)` - calculates - incomplete elliptic integral of the first kind (with `0code>): ` - -- `double ``ROOT::Math::ellint_2(double k,double phi)` - calculates - the complete elliptic integral of the second kind (with `0code>): ` - -- `double ``ROOT::Math::ellint_3(double n,double k,double phi) - `calculates - the complete elliptic integral of the third kind (with `0code>): ` - -- `double ``ROOT::Math::expint(double x)` - calculates the exponential - integral:  - -- `double ``ROOT::Math::hyperg(double a,double b,double c,double x)` - + function: + $$ + N_{\nu} (x) = Y_{\nu} (x) = \left\{ \begin{array}{cl} \frac{J_{\nu} \cos{\nu \pi}-J_{-\nu}(x)}{\sin{\nu \pi}} & \mbox{for non-integral $\nu$} \\ \lim{\mu \to \nu} \frac{J_{\mu} \cos{\mu \pi}-J_{-\mu}(x)}{\sin{\mu \pi}} & \mbox{for integral $\nu$} \end{array} \right. + $$ + +- `double ROOT::Math::ellint_1(double k,double phi)` - calculates + incomplete elliptic integral of the first kind (with $0 \le k^2 \le 1$): + $$ + K(k) = F(k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{\sqrt{1 - k^2 \sin^2{\theta}}} + $$ + +- `double ROOT::Math::ellint_2(double k,double phi)` - calculates + the complete elliptic integral of the second kind (with $0 \le k^2 \le 1$): + $$ + E(k) = E(k , \pi / 2) = \int_{0}^{\pi /2} \sqrt{1 - k^2 \sin^2{\theta}} d \theta + $$ + +- `double ROOT::Math::ellint_3(double n,double k,double phi)` - calculates + the complete elliptic integral of the third kind (with $0 \le k^2 \le 1$): + $$ + \Pi (n, k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{(1 - n \sin^2{\theta})\sqrt{1 - k^2 \sin^2{\theta}}} + $$ + +- `double ROOT::Math::expint(double x)` - calculates the exponential + integral: + $$ + Ei(x) = - \int_{-x}^{\infty} \frac{e^{-t}}{t} dt + $$ + +- `double ROOT::Math::hyperg(double a,double b,double c,double x)` - calculates Gauss' hyper-geometric function: -  - -- `double ``ROOT::Math::legendre(unsigned l,double x)` - calculates - the Legendre polynomials for `l>=0`, - `|x|code> in the Rodrigues representation: ` - -- `double ``ROOT::Math::riemann_zeta(double x)` - calculates the - Riemann zeta function:  - -- `double ``ROOT::Math::sph_bessel(unsigned n,double x)` - calculates + $$ + _{2}F_{1}(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a) \Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} + $$ + +- `double ROOT::Math::legendre(unsigned l,double x)` - calculates + the Legendre polynomials for $l \ge 0$, $|x| \le 1$ in the Rodrigues + representation: + $$ + P_{l}(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l + $$ + +- `double ROOT::Math::riemann_zeta(double x)` - calculates the + Riemann zeta function: + $$ + \zeta (x) = \left\{ \begin{array}{cl} \sum_{k=1}^{\infty}k^{-x} & \mbox{for $x > 1$} \\ 2^x \pi^{x-1} \sin{(\frac{1}{2}\pi x)} \Gamma(1-x) \zeta (1-x) & \mbox{for $x < 1$} \end{array} \right. + $$ + +- `double ROOT::Math::sph_bessel(unsigned n,double x)` - calculates the spherical Bessel functions of the first kind (also called - regular spherical Bessel functions):  + regular spherical Bessel functions): + $$ + j_{n}(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x) + $$ -- `double ``ROOT::Math::sph_neumann(unsigned n,double x)` - calculates +- `double ROOT::Math::sph_neumann(unsigned n,double x)` - calculates the spherical Bessel functions of the second kind (also called irregular spherical Bessel functions or spherical Neumann - functions): + functions): + $$ + n_n(x) = y_n(x) = \sqrt{\frac{\pi}{2x}} N_{n+1/2}(x) + $$ ### Probability Density Functions (PDF) @@ -1514,7 +1612,9 @@ cumulative distribution functions and their complements. The functions with extension `_cdf` calculate the lower tail integral of the probability density function: - +$$ +D(x) = \int_{-\infty}^{x} p(x') dx' +$$ while those with the `cdf_c` extension calculate the upper tail of the probability density function, so-called in statistics the survival @@ -1523,20 +1623,20 @@ function. For example, the function: ``` {.cpp} double ROOT::Math::gaussian_cdf(double x,double sigma,double x0=0); ``` - evaluates the lower tail of the Gaussian distribution: - - +$$ +D(x) = \int_{-\infty}^{x} {1 \over \sqrt{2 \pi \sigma^2}} e^{-(x'-x_0)^2 / 2\sigma^2} dx' +$$ while the function: ``` {.cpp} double ROOT::Math::gaussian_cdf_c(double x, double sigma, double x0=0); ``` - evaluates the upper tail of the Gaussian distribution: - - +$$ +D(x) = \int_{x}^{+\infty} {1 \over \sqrt{2 \pi \sigma^2}} e^{-(x'-x_0)^2 / 2\sigma^2} dx' +$$ The cumulative distributions functions are defined in the header file `Math/ProbFunc.h`. The majority of the CDF's are present in the @@ -1559,8 +1659,7 @@ the header file `Math/ProbFuncInv.h`. The following picture illustrates the available statistical functions (PDF, CDF and quantiles) in the case of the normal distribution. - + ## Linear Algebra: SMatrix Package @@ -1649,9 +1748,10 @@ In the following example we assume that we are using the namespace **`ROOT::Math`** ``` {.cpp} -SVector<double,3> v; //create an empty vector of size 3 ( v[0]=v[1]=v[2]=0) -double d[3] = {1,2,3}; -SVector<double,3> v(d,3); //create a vector from a C array + //create an empty vector of size 3 ( v[0]=v[1]=v[2]=0) + SVector<double,3> v; + double d[3] = {1,2,3}; + SVector<double,3> v(d,3); //create a vector from a C array ``` #### Accessing and Setting Methods @@ -1663,15 +1763,15 @@ performed on the passed index. The full vector elements can be set also by using the SetElements function passing a generic iterator. ``` {.cpp} -double x = m(i); // return the i-th element -x = *(m.begin()+i); // return the i-th element -v[0] = 1; // set the first element -v(1) = 2; // set the second element -*(v.begin()+3) = 3; // set the third element -std::vector<double> w(3); + double x = m(i); // return the i-th element + x = *(m.begin()+i); // return the i-th element + v[0] = 1; // set the first element + v(1) = 2; // set the second element + *(v.begin()+3) = 3; // set the third element + std::vector<double> w(3); -// set vector elements from a std::vector<double>::iterator -v.SetElements(w.begin(),w.end()); + // set vector elements from a std::vector<double>::iterator + v.SetElements(w.begin(),w.end()); ``` In addition there are methods to place a sub-vector in a vector. If the @@ -1679,13 +1779,15 @@ size of the sub-vector is larger than the vector size a static assert (a compilation error) is produced. ``` {.cpp} -SVector>double,N> v; -SVector>double,M> w; -// M <= N otherwise a compilation error is obtained later -// place a vector of size M starting from element ioff, v[ioff+i]=w[i] -v.Place_at(w,ioff); -// return a sub-vector of size M starting from v[ioff]: w[i]=v[ioff+i] -w = v.Sub < SVector>double,M> > (ioff); + SVector>double,N> v; + SVector>double,M> w; + // M <= N otherwise a compilation error is obtained later + // place a vector of size M starting from + // element ioff, v[ioff+i]=w[i] + v.Place_at(w,ioff); + // return a sub-vector of size M starting from + // v[ioff]: w[i]=v[ioff+i] + w = v.Sub < SVector>double,M> > (ioff); ``` For the vector functions see later in the Matrix and Vector Operators @@ -1716,7 +1818,14 @@ template parameters, which define at compile time, its properties: of size `3x3`, the data `{a0,a1,...,a8}` are stored in the following order: --  +$$ +M = + \left(\begin{array}{ccc} + a_0 & a_1 & a_2 \\ + a_3 & a_4 & a_5 \\ + a_6 & a_7 & a_8 + \end{array}\right) +$$ - **`ROOT::Math::MatRepSym`** for a symmetric matrix of size `NxN`. This class is a template on the contained type and on the symmetric @@ -1726,7 +1835,14 @@ template parameters, which define at compile time, its properties: convention. For example for a symmetric `3x3` matrix the order of the `6` independent elements `{a0,a1,...,a5}` is: --  +$$ +M = + \left(\begin{array}{ccc} + a_0 & a_1 & a_3 \\ + a_1 & a_2 & a_4 \\ + a_3 & a_4 & a_5 + \end{array}\right) +$$ #### Creating a Matrix @@ -1762,36 +1878,38 @@ specified. Furthermore, for a general square matrix, the number of column may be as well omitted. ``` {.cpp} -// typedef definitions used in the following declarations -typedef ROOT::Math::SMatrix<double,3> SMatrix33; -typedef ROOT::Math::SMatrix<double,2> SMatrix22; -typedef ROOT::Math::SMatrix<double,3,3, -ROOT::Math::MatRepSym<double,3>> SMatrixSym3; -typedef ROOT::Math::SVector>double,2> SVector2; -typedef ROOT::Math::SVector>double,3> SVector3; -typedef ROOT::Math::SVector>double,6> SVector6; -SMatrix33 m0; // create a zero 3x3 matrix -// create an 3x3 identity matrix -SMatrix33 i = ROOT::Math::SMatrixIdentity(); -double a[9] = {1,2,3,4,5,6,7,8,9}; // input matrix data -// create a matrix using the a[] data -SMatrix33 m(a,9); // this will produce the 3x3 matrix - // ( 1 2 3 ) - // ( 4 5 6 ) - // ( 7 8 9 ) + // typedef definitions used in the following declarations + typedef ROOT::Math::SMatrix<double,3> SMatrix33; + typedef ROOT::Math::SMatrix<double,2> SMatrix22; + typedef ROOT::Math::SMatrix<double,3,3, + ROOT::Math::MatRepSym<double,3>> SMatrixSym3; + typedef ROOT::Math::SVector>double,2> SVector2; + typedef ROOT::Math::SVector>double,3> SVector3; + typedef ROOT::Math::SVector>double,6> SVector6; + SMatrix33 m0; // create a zero 3x3 matrix + // create an 3x3 identity matrix + SMatrix33 i = ROOT::Math::SMatrixIdentity(); + double a[9] = {1,2,3,4,5,6,7,8,9}; // input matrix data + // create a matrix using the a[] data + SMatrix33 m(a,9); // this will produce the 3x3 matrix + // ( 1 2 3 ) + // ( 4 5 6 ) + // ( 7 8 9 ) ``` Example to fill a symmetric matrix from an `std::vector`: ``` {.cpp} -std::vector<double> v(6); -for (int i = 0; i<6; ++i) v[i] = double(i+1); -SMatrixSym3 s(v.begin(),v.end()) // this will produce the symmetric matrix -// ( 1 2 4 ) - // ( 2 3 5 ) - // ( 4 5 6 ) -//create a general matrix from a symmetric matrix (the opposite will not compile) -SMatrix33 m2 = s; + std::vector<double> v(6); + for (int i = 0; i<6; ++i) v[i] = double(i+1); + SMatrixSym3 s(v.begin(),v.end()) // this will produce the + // symmetric matrix + // ( 1 2 4 ) + // ( 2 3 5 ) + // ( 4 5 6 ) + //create a general matrix from a symmetric matrix (the opposite + // will not compile) + SMatrix33 m2 = s; ``` #### Accessing and Setting Methods @@ -1808,18 +1926,18 @@ iterator access methods which behave differently (it follows the data order). ``` {.cpp} -SMatrix33 m; -m(0,0) = 1; // set the element in first row and first column -*(m.begin()+1) = 2; // set the second element (0,1) -double d[9]={1,2,3,4,5,6,7,8,9}; -m.SetElements(d,d+9); // set the d[] values in m -double x = m(2,1); // return the element in 3 -x = m.apply(7); // return the 8-th element (row=2,col=1) -x = *(m.begin()+7); // return the 8-th element (row=2,col=1) -// symmetric matrices -//(note the difference in behavior between apply and the iterators) -x = *(m.begin()+4) // return the element (row=2,col=1) -x = m.apply(7); // returns again the (row=2,col=1) element + SMatrix33 m; + m(0,0) = 1; // set the element in first row and first column + *(m.begin()+1) = 2; // set the second element (0,1) + double d[9]={1,2,3,4,5,6,7,8,9}; + m.SetElements(d,d+9); // set the d[] values in m + double x = m(2,1); // return the element in 3 + x = m.apply(7); // return the 8-th element (row=2,col=1) + x = *(m.begin()+7); // return the 8-th element (row=2,col=1) + // symmetric matrices + //(note the difference in behavior between apply and the iterators) + x = *(m.begin()+4) // return the element (row=2,col=1) + x = m.apply(7); // returns again the (row=2,col=1) element ``` There are methods to place and/or retrieve **`ROOT::Math::SVector`** @@ -1830,42 +1948,48 @@ matrix size a static assert (a compilation error) is produced. The non-const methods are: ``` {.cpp} -SMatrix33 m; -SVector2 v2(1,2); -// place a vector in the first row from element (0,1) : m(0,1)=v2[0] -m.Place_in_row(v2,0,1); -// place the vector in the second column from (0,1) : m(0,1) = v2[0] -m.Place in_col(v2,0,1); -SMatrix22 m2; -// place m2 in m starting from the element (1,1) : m(1,1) = m2(0,0) -m.Place_at(m2,1,1); -SVector3 v3(1,2,3); -// set v3 as the diagonal elements of m : m(i,i) = v3[i] for i=0,1,2 -m.SetDiagonal(v3) + SMatrix33 m; + SVector2 v2(1,2); + // place a vector in the first row from + // element (0,1) : m(0,1)=v2[0] + m.Place_in_row(v2,0,1); + // place the vector in the second column from + // (0,1) : m(0,1) = v2[0] + m.Place in_col(v2,0,1); + SMatrix22 m2; + // place m2 in m starting from the + // element (1,1) : m(1,1) = m2(0,0) + m.Place_at(m2,1,1); + SVector3 v3(1,2,3); + // set v3 as the diagonal elements + // of m : m(i,i) = v3[i] for i=0,1,2 + m.SetDiagonal(v3) ``` The const methods retrieving contents (getting slices of a matrix) are: ``` {.cpp} -a = {1,2,3,4,5,6,7,8,9}; -SMatrix33 m(a,a+9); -SVector3 irow = m.Row(0); // return as vector the first row -SVector3 jcol = m.Col(1); // return as vector the second column + a = {1,2,3,4,5,6,7,8,9}; + SMatrix33 m(a,a+9); + SVector3 irow = m.Row(0); // return as vector the first row + SVector3 jcol = m.Col(1); // return as vector the second column -// return a slice of the first row from (0,1): r2[0]= m(0,1); r2[1]=m(0,2) -SVector2 r2 = m.SubRow<SVector2> (0,1); -// return a slice of the second column from (0,1): c2[0] = m(0,1); c2[1] = m(1,1) -SVector2 c2 = m.SubCol<SVector2> (1,0); + // return a slice of the first row from + // (0,1): r2[0]= m(0,1); r2[1]=m(0,2) + SVector2 r2 = m.SubRow<SVector2> (0,1); + // return a slice of the second column from + // (0,1): c2[0] = m(0,1); c2[1] = m(1,1) + SVector2 c2 = m.SubCol<SVector2> (1,0); -// return a sub-matrix 2x2 with the upper left corner at(1,1) -SMatrix22 subM = m.Sub<SMatrix22> (1,1); + // return a sub-matrix 2x2 with the upper left corner at(1,1) + SMatrix22 subM = m.Sub<SMatrix22> (1,1); -// return the diagonal element in a SVector -SVector3 diag = m.Diagonal(); + // return the diagonal element in a SVector + SVector3 diag = m.Diagonal(); -// return the upper(lower) block of the matrix m -SVector6 vub = m.UpperBlock(); // vub = [ 1, 2, 3, 5, 6, 9 ] -SVector6 vlb = m.LowerBlock(); // vlb = [ 1, 4, 5, 7, 8, 9 ] + // return the upper(lower) block of the matrix m + SVector6 vub = m.UpperBlock(); // vub = [ 1, 2, 3, 5, 6, 9 ] + SVector6 vlb = m.LowerBlock(); // vlb = [ 1, 4, 5, 7, 8, 9 ] ``` #### Linear Algebra Matrix Functions (Inversion, Determinant) @@ -1880,21 +2004,24 @@ used while for a large `(N>6)` general matrix an LU factorization is performed using the same algorithm as in the CERNLIB routine `dinv`. ``` {.cpp} -// Invert a NxN matrix. -// The inverted matrix replaces the existing one if the result is successful -bool ret = m.Invert(); // return the inverse matrix of m. + // Invert a NxN matrix. + // The inverted matrix replaces the existing one if the + // result is successful + bool ret = m.Invert(); // return the inverse matrix of m. -// If the inversion fails ifail is different than zero ??? -int ifail = 0; -ifail = m.Inverse(ifail); + // If the inversion fails ifail is different than zero ??? + int ifail = 0; + ifail = m.Inverse(ifail); -// determinant of a square matrix - calculate the determinant modyfing the -// matrix content and returns it if the calculation was successful -double det; -bool ret = m.Det(det); + // determinant of a square matrix - calculate the determinant + // modyfing the matrix content and returns it if the calculation + // was successful + double det; + bool ret = m.Det(det); -// calculate determinant by using a temporary matrix; preserves matrix content -bool ret = n.Det2(det); + // calculate determinant by using a temporary matrix; preserves + // matrix content + bool ret = n.Det2(det); ``` ### Example: Matrix and Vector Functions and Operators @@ -1908,21 +2035,26 @@ vectors or matrices of the same type (and size) and `a` is a scalar value: ``` {.cpp} -m1 == m2 //returns whether m1 is equal to m2 (element by element comparison) -m1 != m2 //returns whether m1 is NOT equal to m2 (element by element comparison) -m1 < m2 //returns whether m1 is less than m2 (element wise comparison) -m1 > m2 //returns whether m1 is greater than m2 (element wise comparison) - -//in the following m1 and m3 can be general and m2 symmetric, but not vice-versa + m1 == m2 // returns whether m1 is equal to m2 + // (element by element comparison) + m1 != m2 // returns whether m1 is NOT equal to m2 + // (element by element comparison) + m1 < m2 // returns whether m1 is less than m2 + // (element wise comparison) + m1 > m2 // returns whether m1 is greater than m2 + // (element wise comparison) -m1 += m2 // add m2 to m1 -m1 -= m2 // subtract m2 to m1 -m3 = m1 + m2 // addition -m1 - m2 // subtraction + // in the following m1 and m3 can be general and m2 symmetric, + // but not vice-versa -// Multiplication and division via a scalar value a + m1 += m2 // add m2 to m1 + m1 -= m2 // subtract m2 to m1 + m3 = m1 + m2 // addition + m1 - m2 // subtraction -m3 = a*m1; m3 = m1*a; m3 = m1/a; + // Multiplication and division via a scalar value a + + m3 = a*m1; m3 = m1*a; m3 = m1/a; ``` **Vector-Vector multiplication:** The operator `*` defines an element by @@ -1934,23 +2066,21 @@ for vector sizes of 3), `ROOT::Math::`**`Cross`, and the Tensor product, **Matrix - Vector multiplication:** The operator `*` defines the matrix-vector multiplication: - -. The operation compiles only if the matrix +$y_i = \sum_j M_{i,j} x_j$. The operation compiles only if the matrix and the vectors have the right sizes. ``` {.cpp} -// M is a N1xN2 matrix, x is a N2 size vector, y is a N1 size vector -y = M * x + //M is a N1xN2 matrix, x is a N2 size vector, y is a N1 size vector + y = M * x ``` **Matrix - Matrix multiplication:** The operator `*` defines the matrix-matrix multiplication: - -. +$C_{i,j} = \sum_k A_{i,k} B_{k,j}$. ``` {.cpp} -// A is a N1xN2 matrix, B is a N2xN3 matrix and C is a N1xN3 matrix -C = A * B + // A is a N1xN2 matrix, B is a N2xN3 matrix and C is a N1xN3 matrix + C = A * B ``` The operation compiles only if the matrices have the right size. In the @@ -2000,9 +2130,9 @@ One can print (or write in an output stream) Vectors and Matrices) using the `Print` method or the `<<` operator: ``` {.cpp} -// m is a SMatrix or a SVector object -m.Print(std::cout); -std::cout << m << std::endl; + // m is a SMatrix or a SVector object + m.Print(std::cout); + std::cout << m << std::endl; ``` In the ROOT distribution, the CINT dictionary is generated for `SMatrix` @@ -2039,15 +2169,16 @@ plug-in. For example for using it in histogram fitting, one only needs to do: ``` {.cpp} -TVirtualFitter::SetDefaultFitter("Minuit2"); //or Fumili2 for the FUMILI algorithmhistogram->Fit(); + TVirtualFitter::SetDefaultFitter("Minuit2"); //or Fumili2 for the + // FUMILI algorithmhistogram->Fit(); ``` For minimization problem, providing an FCN function to minimize, one can do: ``` {.cpp} -TVirtualFitter::SetDefaultFitter("Minuit2"); -TVirtualFitter * minuit2 = TVirtualFitter::Fitter(0,2); + TVirtualFitter::SetDefaultFitter("Minuit2"); + TVirtualFitter * minuit2 = TVirtualFitter::Fitter(0,2); ``` Then set the parameters, the FCN and minimize using the @@ -2058,7 +2189,7 @@ interface. In this case one must use directly the **`TFitterMinuit`** class via the method `SetMinuitFCN`. Examples on how to use the `Minuit2` and `Fumili2` plug-ins are provided -in the tutorials' directory \$ROOTSYS/`tutorials/fit`: +in the tutorials' directory `$ROOTSYS/tutorials/fit`: `minuit2FitBench.C`, `minuit2FitBench2D.C` and `minuit2GausFit.C`. 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