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Documentation

intlinprog

Mixed-integer linear programming (MILP)

Description

Mixed-integer linear programming solver.

Finds the minimum of a problem specified by

f, x, intcon, b, beq, lb, and ub are vectors, and A and Aeq are matrices.

You can specify f, intcon, lb, and ub as vectors or arrays. See Matrix Arguments.

Note

intlinprog applies only to the solver-based approach. For a discussion of the two optimization approaches, see First Choose Problem-Based or Solver-Based Approach.

example

x = intlinprog(f,intcon,A,b) solves min f'*x such that the components of x in intcon are integers, and A*x ≤ b.

x = intlinprog(f,intcon,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. Set A = [] and b = [] if no inequalities exist.

example

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb ≤ x ≤ ub. Set Aeq = [] and beq = [] if no equalities exist.

example

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,x0)optimizes using an initial feasible point x0. Set lb = [] and ub = [] if no bounds exist.

example

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,x0,options) minimizes using the optimization options specified in options. Use optimoptions to set these options. Set x0 = [] if no initial point exists.

example

x = intlinprog(problem) uses a problem structure to encapsulate all solver inputs. You can import a problem structure from an MPS file using mpsread. You can also create a problem structure from an OptimizationProblem object by using prob2struct.

example

[x,fval,exitflag,output] = intlinprog(___), for any input arguments described above, returns fval = f'*x, a value exitflag describing the exit condition, and a structure output containing information about the optimization process.

Examples

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Solve the problem

$\underset{x}{\mathrm{min}}8{x}_{1}+{x}_{2}\phantom{\rule{0.5em}{0ex}}subject\phantom{\rule{0.5em}{0ex}}to\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}{x}_{2}\phantom{\rule{0.5em}{0ex}}is\phantom{\rule{0.5em}{0ex}}an\phantom{\rule{0.5em}{0ex}}integer\\ {x}_{1}+2{x}_{2}\ge -14\\ -4{x}_{1}-{x}_{2}\le -33\\ 2{x}_{1}+{x}_{2}\le 20.\end{array}$

Write the objective function vector and vector of integer variables.

f = [8;1];
intcon = 2;

Convert all inequalities into the form A*x <= b by multiplying “greater than” inequalities by -1.

A = [-1,-2;
-4,-1;
2,1];
b = [14;-33;20];

Call intlinprog.

x = intlinprog(f,intcon,A,b)
LP:                Optimal objective value is 59.000000.

Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap
tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default
value). The intcon variables are integer within tolerance,
options.IntegerTolerance = 1e-05 (the default value).
x = 2×1

6.5000
7.0000

Solve the problem

$\underset{x}{\mathrm{min}}\left(-3{x}_{1}-2{x}_{2}-{x}_{3}\right)\phantom{\rule{0.5em}{0ex}}subject\phantom{\rule{0.5em}{0ex}}to\left\{\begin{array}{l}{x}_{3}\phantom{\rule{0.5em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12.\end{array}$

Write the objective function vector and vector of integer variables.

f = [-3;-2;-1];
intcon = 3;

Write the linear inequality constraints.

A = [1,1,1];
b = 7;

Write the linear equality constraints.

Aeq = [4,2,1];
beq = 12;

Write the bound constraints.

lb = zeros(3,1);
ub = [Inf;Inf;1]; % Enforces x(3) is binary

Call intlinprog.

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)
LP:                Optimal objective value is -12.000000.

Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap
tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default
value). The intcon variables are integer within tolerance,
options.IntegerTolerance = 1e-05 (the default value).
x = 3×1

0
5.5000
1.0000

Compare the number of steps to solve an integer programming problem both with and without an initial feasible point. The problem has eight variables, four linear equality constraints, and has all variables restricted to be positive.

Define the linear equality constraint matrix and vector.

Aeq = [22    13    26    33    21     3    14    26
39    16    22    28    26    30    23    24
18    14    29    27    30    38    26    26
41    26    28    36    18    38    16    26];
beq = [ 7872
10466
11322
12058];

Set lower bounds that restrict all variables to be nonnegative.

N = 8;
lb = zeros(N,1);

Specify that all variables are integer-valued.

intcon = 1:N;

Set the objective function vector f.

f = [2    10    13    17     7     5     7     3];

Solve the problem without using an initial point, and examine the display to see the number of branch-and-bound nodes.

[x1,fval1,exitflag1,output1] = intlinprog(f,intcon,[],[],Aeq,beq,lb);
LP:                Optimal objective value is 1554.047531.

Cut Generation:    Applied 8 strong CG cuts.
Lower bound is 1591.000000.

Branch and Bound:

nodes     total   num int        integer       relative
explored  time (s)  solution           fval        gap (%)
10000      1.03         0              -              -
18027      1.69         1   2.906000e+03   4.509804e+01
21859      2.12         2   2.073000e+03   2.270974e+01
23546      2.30         3   1.854000e+03   1.180593e+01
24121      2.36         3   1.854000e+03   1.563342e+00
24294      2.39         3   1.854000e+03   0.000000e+00

Optimal solution found.

Intlinprog stopped because the objective value is within a gap tolerance of the
optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon
variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the
default value).

For comparison, find the solution using an initial feasible point.

x0 = [8 62 23 103 53 84 46 34];
[x2,fval2,exitflag2,output2] = intlinprog(f,intcon,[],[],Aeq,beq,lb,[],x0);
LP:                Optimal objective value is 1554.047531.

Cut Generation:    Applied 8 strong CG cuts.
Lower bound is 1591.000000.
Relative gap is 59.20%.

Branch and Bound:

nodes     total   num int        integer       relative
explored  time (s)  solution           fval        gap (%)
3627      0.48         2   2.154000e+03   2.593968e+01
5844      0.70         3   1.854000e+03   1.180593e+01
6204      0.73         3   1.854000e+03   1.455526e+00
6400      0.77         3   1.854000e+03   0.000000e+00

Optimal solution found.

Intlinprog stopped because the objective value is within a gap tolerance of the
optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon
variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the
default value).
• Without an initial point, intlinprog took about 30,000 branch-and-bound steps.

• Using an initial point, intlinprog took about 5,000 steps.

Giving an initial point does not always help. For this problem, giving an initial point saves time and computational steps. However, for some problems, giving an initial point can cause intlinprog to take more steps.

Solve the problem

$\underset{x}{\mathrm{min}}\left(-3{x}_{1}-2{x}_{2}-{x}_{3}\right)\phantom{\rule{0.5em}{0ex}}subject\phantom{\rule{0.5em}{0ex}}to\left\{\begin{array}{l}{x}_{3}\phantom{\rule{0.5em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12\end{array}$

without showing iterative display.

Specify the solver inputs.

f = [-3;-2;-1];
intcon = 3;
A = [1,1,1];
b = 7;
Aeq = [4,2,1];
beq = 12;
lb = zeros(3,1);
ub = [Inf;Inf;1]; % enforces x(3) is binary
x0 = [];

Specify no display.

options = optimoptions('intlinprog','Display','off');

Run the solver.

x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub,x0,options)
x = 3×1

0
5.5000
1.0000

This example shows how to set up a problem using the problem-based approach and then solve it using the solver-based approach. The problem is

$\underset{x}{\mathrm{min}}\left(-3{x}_{1}-2{x}_{2}-{x}_{3}\right)\phantom{\rule{0.5em}{0ex}}subject\phantom{\rule{0.5em}{0ex}}to\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{l}{x}_{3}\phantom{\rule{0.5em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12\end{array}$

Create an OptimizationProblem object named prob to represent this problem. To specify a binary variable, create an optimization variable with integer type, a lower bound of 0, and an upper bound of 1.

x = optimvar('x',2,'LowerBound',0);
xb = optimvar('xb','LowerBound',0,'UpperBound',1,'Type','integer');
prob = optimproblem('Objective',-3*x(1)-2*x(2)-xb);
cons1 = sum(x) + xb <= 7;
cons2 = 4*x(1) + 2*x(2) + xb == 12;
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;

Convert the problem object to a problem structure.

problem = prob2struct(prob);

Solve the resulting problem structure.

[sol,fval,exitflag,output] = intlinprog(problem)
LP:                Optimal objective value is -12.000000.

Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap
tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default
value). The intcon variables are integer within tolerance,
options.IntegerTolerance = 1e-05 (the default value).
sol = 3×1

0
5.5000
1.0000

fval = -12
exitflag = 1
output = struct with fields:
relativegap: 0
absolutegap: 0
numfeaspoints: 1
numnodes: 0
constrviolation: 0
message: 'Optimal solution found....'

Both sol(1) and sol(3) are binary-valued. Which value corresponds to the binary optimization variable xb?

prob.Variables
ans = struct with fields:
x: [2x1 optim.problemdef.OptimizationVariable]
xb: [1x1 optim.problemdef.OptimizationVariable]

The variable xb appears last in the Variables display, so xb corresponds to sol(3) = 1. See Algorithms.

Call intlinprog with more outputs to see solution details and process.

The goal is to solve the problem

$\underset{x}{\mathrm{min}}\left(-3{x}_{1}-2{x}_{2}-{x}_{3}\right)\phantom{\rule{0.5em}{0ex}}subject\phantom{\rule{0.5em}{0ex}}to\left\{\begin{array}{l}{x}_{3}\phantom{\rule{0.5em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12.\end{array}$

Specify the solver inputs.

f = [-3;-2;-1];
intcon = 3;
A = [1,1,1];
b = 7;
Aeq = [4,2,1];
beq = 12;
lb = zeros(3,1);
ub = [Inf;Inf;1]; % enforces x(3) is binary

Call intlinprog with all outputs.

[x,fval,exitflag,output] = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)
LP:                Optimal objective value is -12.000000.

Optimal solution found.

Intlinprog stopped at the root node because the objective value is within a gap
tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default
value). The intcon variables are integer within tolerance,
options.IntegerTolerance = 1e-05 (the default value).
x = 3×1

0
5.5000
1.0000

fval = -12
exitflag = 1
output = struct with fields:
relativegap: 0
absolutegap: 0
numfeaspoints: 1
numnodes: 0
constrviolation: 0
message: 'Optimal solution found....'

The output structure shows numnodes is 0. This means intlinprog solved the problem before branching. This is one indication that the result is reliable. Also, the absolutegap and relativegap fields are 0. This is another indication that the result is reliable.

Input Arguments

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Coefficient vector, specified as a real vector or real array. The coefficient vector represents the objective function f'*x. The notation assumes that f is a column vector, but you are free to use a row vector or array. Internally, linprog converts f to the column vector f(:).

If you specify f = [], intlinprog tries to find a feasible point without trying to minimize an objective function.

Example: f = [4;2;-1.7];

Data Types: double

Vector of integer constraints, specified as a vector of positive integers. The values in intcon indicate the components of the decision variable x that are integer-valued. intcon has values from 1 through numel(f).

intcon can also be an array. Internally, intlinprog converts an array intcon to the vector intcon(:).

Example: intcon = [1,2,7] means x(1), x(2), and x(7) take only integer values.

Data Types: double

Linear inequality constraint matrix, specified as a matrix of doubles. A represents the linear coefficients in the constraints A*x  b. A has size M-by-N, where M is the number of constraints and N = numel(f). To save memory, A can be sparse.

Example: A = [4,3;2,0;4,-1]; means three linear inequalities (three rows) for two decision variables (two columns).

Data Types: double

Linear inequality constraint vector, specified as a vector of doubles. b represents the constant vector in the constraints A*x  b. b has length M, where A is M-by-N.

Example: [4,0]

Data Types: double

Linear equality constraint matrix, specified as a matrix of doubles. Aeq represents the linear coefficients in the constraints Aeq*x = beq. Aeq has size Meq-by-N, where Meq is the number of constraints and N = numel(f). To save memory, Aeq can be sparse.

Example: A = [4,3;2,0;4,-1]; means three linear inequalities (three rows) for two decision variables (two columns).

Data Types: double

Linear equality constraint vector, specified as a vector of doubles. beq represents the constant vector in the constraints Aeq*x = beq. beq has length Meq, where Aeq is Meq-by-N.

Example: [4,0]

Data Types: double

Lower bounds, specified as a vector or array of doubles. lb represents the lower bounds elementwise in lb  x  ub.

Internally, intlinprog converts an array lb to the vector lb(:).

Example: lb = [0;-Inf;4] means x(1) ≥ 0, x(3) ≥ 4.

Data Types: double

Upper bounds, specified as a vector or array of doubles. ub represents the upper bounds elementwise in lb  x  ub.

Internally, intlinprog converts an array ub to the vector ub(:).

Example: ub = [Inf;4;10] means x(2) ≤ 4, x(3) ≤ 10.

Data Types: double

Initial point, specified as a real array. The number of elements in x0 is the same as the number of elements of f, when f exists. Otherwise, the number is the same as the number of columns of A or Aeq. Internally, the solver converts an array x0 into a vector x0(:).

Providing x0 can change the amount of time intlinprog takes to converge. It is difficult to predict how x0 affects the solver. For suggestions on using appropriate Heuristics with x0, see Tips.

x0 must be feasible with respect to all constraints. If x0 is not feasible, the solver errors. If you do not have a feasible x0, set x0 = [].

Example: x0 = 100*rand(size(f))

Data Types: double

Options for intlinprog, specified as the output of optimoptions.

Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Options.

OptionDescriptionDefault
AbsoluteGapTolerance

Nonnegative real. intlinprog stops if the difference between the internally calculated upper (U) and lower (L) bounds on the objective function is less than or equal to AbsoluteGapTolerance:

U – L <= AbsoluteGapTolerance.

0
BranchRule

Rule for choosing the component for branching:

• 'maxpscost' — The fractional component with maximum pseudocost. See Branch and Bound.

• 'strongpscost' — The fractional component with maximum pseudocost and a more accurate estimate of pseudocost than in 'maxpscost'. See Branch and Bound.

• 'reliability' — The fractional component with maximum pseudocost and an even more accurate estimate of pseudocost than in 'strongpscost'. See Branch and Bound.

• 'mostfractional' — The component whose fractional part is closest to 1/2.

• 'maxfun' — The fractional component with a maximal corresponding component in the absolute value of the objective vector f.

'reliability'
ConstraintToleranceReal from 1e-9 through 1e-3 that is the maximum discrepancy that linear constraints can have and still be considered satisfied. ConstraintTolerance is not a stopping criterion.1e-4
CutGeneration

Level of cut generation (see Cut Generation):

• 'none' — No cuts. Makes CutMaxIterations irrelevant.

• 'basic' — Normal cut generation.

• 'intermediate' — Use more cut types.

• 'advanced' — Use most cut types.

'basic'
CutMaxIterationsNumber of passes through all cut generation methods before entering the branch-and-bound phase, an integer from 1 through 50. Disable cut generation by setting the CutGeneration option to 'none'.10
Display

Level of display (see Iterative Display):

• 'off' or 'none' — No iterative display

• 'final' — Show final values only

• 'iter' — Show iterative display

'iter'
Heuristics

Algorithm for searching for feasible points (see Heuristics for Finding Feasible Solutions):

• 'basic'

• 'intermediate'

• 'rins'

• 'round'

• 'diving'

• 'rins-diving'

• 'round-diving'

• 'none'

'basic'
HeuristicsMaxNodesStrictly positive integer that bounds the number of nodes intlinprog can explore in its branch-and-bound search for feasible points. Applies only to 'rss' and 'rins'. See Heuristics for Finding Feasible Solutions.50
IntegerPreprocess

Types of integer preprocessing (see Mixed-Integer Program Preprocessing):

• 'none' — Use very few integer preprocessing steps.

• 'basic' — Use a moderate number of integer preprocessing steps.

• 'advanced' — Use all available integer preprocessing steps.

'basic'
IntegerToleranceReal from 1e-6 through 1e-3, where the maximum deviation from integer that a component of the solution x can have and still be considered an integer. IntegerTolerance is not a stopping criterion.1e-5
LPMaxIterationsStrictly positive integer, the maximum number of simplex algorithm iterations per node during the branch-and-bound process.

max(3e4, 10*(numberOfEqualities + numberOfInequalities + numberOfVariables))

In this expression, numberOfEqualities means the number of rows of Aeq, numberOfInequalities means the number of rows of A, and numberOfVariables means the number of elements of f.

LPOptimalityToleranceNonnegative real where reduced costs must exceed LPOptimalityTolerance for a variable to be taken into the basis.1e-7
LPPreprocess

Type of preprocessing for the solution to the relaxed linear program (see Linear Program Preprocessing):

• 'none' — No preprocessing.

• 'basic' — Use preprocessing.

'basic'
MaxNodesStrictly positive integer that is the maximum number of nodes intlinprog explores in its branch-and-bound process.1e7
MaxFeasiblePointsStrictly positive integer. intlinprog stops if it finds MaxFeasiblePoints integer feasible points.Inf
MaxTimePositive real that is the maximum time in seconds that intlinprog runs.7200
NodeSelection

Choose the node to explore next.

• 'simplebestproj' — Best projection. See Branch and Bound.

• 'minobj' — Explore the node with the minimum objective function.

• 'mininfeas' — Explore the node with the minimal sum of integer infeasibilities. See Branch and Bound.

'simplebestproj'
ObjectiveCutOffReal greater than -Inf. During the branch-and-bound calculation, intlinprog discards any node where the linear programming solution has an objective value exceeding ObjectiveCutOff.Inf
ObjectiveImprovementThresholdNonnegative real. intlinprog changes the current feasible solution only when it locates another with an objective function value that is at least ObjectiveImprovementThreshold lower: (fold – fnew)/(1 + |fold|) > ObjectiveImprovementThreshold.0
OutputFcn

Specify one or more functions that an optimization function calls at events as 'savemilpsolutions', a function handle, or a cell array of function handles. For custom output functions, pass function handles.

• 'savemilpsolutions' collects the integer-feasible points in the xIntSol matrix in your workspace, where each column is one integer feasible point.

For information on writing a custom output function, see intlinprog Output Function and Plot Function Syntax.

[]
PlotFcn

Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass 'optimplotmilp', a function handle, or a cell array of function handles. For custom plot functions, pass function handles. The default is none ([]):

• 'optimplotmilp' plots the internally-calculated upper and lower bounds on the objective value of the solution.

For information on writing a custom plot function, see intlinprog Output Function and Plot Function Syntax.

[]
RelativeGapTolerance

Real from 0 through 1. intlinprog stops if the relative difference between the internally calculated upper (U) and lower (L) bounds on the objective function is less than or equal to RelativeGapTolerance:

(U – L) / (abs(U) + 1) <= RelativeGapTolerance.

intlinprog automatically modifies the tolerance for large L magnitudes:

tolerance = min(1/(1+|L|), RelativeGapTolerance)

Note

Although you specify RelativeGapTolerance as a decimal number, the iterative display and output.relativegap report the gap as a percentage, meaning 100 times the measured relative gap. If the exit message refers to the relative gap, this value is the measured relative gap, not a percentage.

1e-4
RootLPAlgorithm

Algorithm for solving linear programs:

• 'dual-simplex' — Dual simplex algorithm

• 'primal-simplex' — Primal simplex algorithm

'dual-simplex'
RootLPMaxIterationsNonnegative integer that is the maximum number of simplex algorithm iterations to solve the initial linear programming problem.

max(3e4, 10*(numberOfEqualities + numberOfInequalities + numberOfVariables))

In this expression, numberOfEqualities means the number of rows of Aeq, numberOfInequalities means the number of rows of A, and numberOfVariables means the number of elements of f.

Example: options = optimoptions('intlinprog','MaxTime',120)

Structure encapsulating the inputs and options, specified with the following fields.

 f Vector representing objective f'*x (required) intcon Vector indicating variables that take integer values (required) Aineq Matrix in linear inequality constraints Aineq*x ≤ bineq bineq Vector in linear inequality constraints Aineq*x ≤ bineq Aeq Matrix in linear equality constraints Aeq*x = beq beq Vector in linear equality constraints Aeq*x = beq lb Vector of lower bounds ub Vector of upper bounds x0 Initial feasible point solver 'intlinprog' (required) options Options created using optimoptions (required)

You must specify at least these fields in the problem structure. Other fields are optional:

• f

• intcon

• solver

• options

Example: problem.f = [1,2,3];
problem.intcon = [2,3];
problem.options = optimoptions('intlinprog');
problem.Aineq = [-3,-2,-1];
problem.bineq = -20;
problem.lb = [-6.1,-1.2,7.3];
problem.solver = 'intlinprog';

Data Types: struct

Output Arguments

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Solution, returned as a vector that minimizes f'*x subject to all bounds, integer constraints, and linear constraints.

When a problem is infeasible or unbounded, x is [].

Objective value, returned as the scalar value f'*x at the solution x.

When a problem is infeasible or unbounded, fval is [].

Algorithm stopping condition, returned as an integer identifying the reason the algorithm stopped. The following lists the values of exitflag and the corresponding reasons intlinprog stopped.

 3 The solution is feasible with respect to the relative ConstraintTolerance tolerance, but is not feasible with respect to the absolute tolerance. 2 intlinprog stopped prematurely. Integer feasible point found. 1 intlinprog converged to the solution x. 0 intlinprog stopped prematurely. No integer feasible point found. -1 intlinprog stopped by an output function or plot function. -2 No feasible point found. -3 Root LP problem is unbounded. -9 Solver lost feasibility.

The exit message can give more detailed information on the reason intlinprog stopped, such as exceeding a tolerance.

Exitflags 3 and -9 relate to solutions that have large infeasibilities. These usually arise from linear constraint matrices that have large condition number, or problems that have large solution components. To correct these issues, try to scale the coefficient matrices, eliminate redundant linear constraints, or give tighter bounds on the variables.

Solution process summary, returned as a structure containing information about the optimization process.

relativegap

Relative percentage difference between upper (U) and lower (L) bounds of the objective function that intlinprog calculates in its branch-and-bound algorithm.

relativegap = 100*(U - L) / (abs(U) + 1)

If intcon = [], relativegap = [].

Note

Although you specify RelativeGapTolerance as a decimal number, the iterative display and output.relativegap report the gap as a percentage, meaning 100 times the measured relative gap. If the exit message refers to the relative gap, this value is the measured relative gap, not a percentage.

absolutegap

Difference between upper and lower bounds of the objective function that intlinprog calculates in its branch-and-bound algorithm.

If intcon = [], absolutegap = [].

numfeaspoints

Number of integer feasible points found.

If intcon = [], numfeaspoints = []. Also, if the initial relaxed problem is infeasible, numfeaspoints = [].

numnodes

Number of nodes in branch-and-bound algorithm. If the solution was found during preprocessing or during the initial cuts, numnodes = 0.

If intcon = [], numnodes = [].

constrviolation

Constraint violation that is positive for violated constraints.

constrviolation = max([0; norm(Aeq*x-beq, inf); (lb-x); (x-ub); (Ai*x-bi)])

message

Exit message.

Limitations

• Often, some supposedly integer-valued components of the solution x(intCon) are not precisely integers. intlinprog deems as integers all solution values within IntegerTolerance of an integer.

To round all supposed integers to be exactly integers, use the round function.

x(intcon) = round(x(intcon));

Caution

Rounding solutions can cause the solution to become infeasible. Check feasibility after rounding:

max(A*x - b) % See if entries are not too positive, so have small infeasibility
max(abs(Aeq*x - beq)) % See if entries are near enough to zero
max(x - ub) % Positive entries are violated bounds
max(lb - x) % Positive entries are violated bounds
• intlinprog does not enforce that solution components be integer-valued when their absolute values exceed 2.1e9. When your solution has such components, intlinprog warns you. If you receive this warning, check the solution to see whether supposedly integer-valued components of the solution are close to integers.

• intlinprog does not allow components of the problem, such as coefficients in f, A, or ub, to exceed 1e25 in absolute value. If you try to run intlinprog with such a problem, intlinprog issues an error.

• Currently, you cannot run intlinprog in the Optimization app.

Tips

• To specify binary variables, set the variables to be integers in intcon, and give them lower bounds of 0 and upper bounds of 1.

• Save memory by specifying sparse linear constraint matrices A and Aeq. However, you cannot use sparse matrices for b and beq.

• If you include an x0 argument, intlinprog uses that value in the 'rins' and guided diving heuristics until it finds a better integer-feasible point. So when you provide x0, you can obtain good results by setting the 'Heuristics' option to 'rins-diving' or another setting that uses 'rins'.

• To provide logical indices for integer components, meaning a binary vector with 1 indicating an integer, convert to intcon form using find. For example,

logicalindices = [1,0,0,1,1,0,0];
intcon = find(logicalindices)
intcon =

1     4     5
• intlinprog replaces bintprog. To update old bintprog code to use intlinprog, make the following changes:

• Set intcon to 1:numVars, where numVars is the number of variables in your problem.

• Set lb to zeros(numVars,1).

• Set ub to ones(numVars,1).

• Update any relevant options. Use optimoptions to create options for intlinprog.

• Change your call to bintprog as follows:

[x,fval,exitflag,output] = bintprog(f,A,b,Aeq,Beq,x0,options)
% Change your call to:
[x,fval,exitflag,output] = intlinprog(f,intcon,A,b,Aeq,Beq,lb,ub,x0,options)

Compatibility Considerations

expand all

Behavior changed in R2019a


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# College Math Teaching

## July 30, 2012

### Good writing is difficult! So let’s drop writing requirements from the curriculum….

Filed under: basic algebra, editorial, elementary mathematics — collegemathteaching @ 8:02 pm

Ok, no one I know is advocating for this. But when it comes to basic algebra (the “solve for x” $x^2 + 8x + 12 = 0$ kind of algebra), well:

A TYPICAL American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

Go ahead and read the rest of the article; it is full of the usual “well people aren’t going to use the quadratic formula in real life”, blah, blah, blah.

Fortunately, the reader recommended comments are good.

So, why should algebra remain part of the curriculum? Here is my opinion:

while algebra is a huge human intellectual achievement, it is also a gateway to things like calculus, economics, statistics, chemistry, physical science and other subjects. NOT requiring algebra will intellectually cripple thousands of students right off the bat.

Of course, I disagree with algebra being used as a “capstone” type course in its current form; its current form makes it into a gateway type of course.

One could devise some sort of capstone type algebra course if one wanted to, but it would be different than the preparatory course.

Interestingly enough, when this subject comes up, I usually hear comments of the following type: “well, *I* am very, very smart but I struggled with it, therefore it is unnecessary”….and usually the sole credential for the person’s intellectual ability is, well, their own opinion of said ability. 🙂

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# Geary's C

Geary's C is a measure of spatial autocorrelation. Like temporal autocorrelation, spatial autocorrelation means that adjacent observations of the same phenomenon are correlated. However, temporal autocorrelation is about proximity in time, while spatial autocorrelation is about proximity in (multi-dimensional) space. Spatial autocorrelation is more complex than autocorrelation because the correlation is multi-dimensional and bi-directional.

Geary's C is defined as

$C = \frac{(N-1) \sum_{i} \sum_{j} w_{ij} (X_i-X_j)^2}{2 W \sum_{i}(X_i-\bar X)^2}$

where $N$ is the number of spatial units indexed by $i$ and $j$; $X$ is the variable of interest; $\bar X$ is the mean of $X$; $w_{ij}$ is a matrix of spatial weights; and $W$ is the sum of all $w_{ij}$.

The value of Geary's C lies between 0 and 2. 1 means no spatial autocorrelation. Values lower than 1 demonstrate increasing positive spatial autocorrelation, whilst values higher than 1 illustrate increasing negative spatial autocorrelation.

Geary's C is inversely related to Moran's I, but it is not identical. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation.

Geary's C is also known as Geary's contiguity ratio, Geary's ratio, or the Geary index.[citation needed]

This statistic was developed by Roy C. Geary.[1]

## Sources

1. ^ Geary, R. C. (1954). "The Contiguity Ratio and Statistical Mapping". The Incorporated Statistician (The Incorporated Statistician) 5 (3): 115–145. doi:10.2307/2986645. JSTOR 2986645.


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# Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle

## Theorem

In the words of Euclid:

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.

## Proof

Let $A$ be a solid angle which is contained by the three plane angles $\angle BAC$, $\angle CAD$ and $\angle DAB$.

It is to be demonstrated that any two of these plane angles together are greater than the third.

Suppose $\angle BAC$, $\angle CAD$ and $\angle DAB$ are all equal.

Then any two together are greater than the remaining one and the result follows.

Suppose otherwise.

Without loss of generality, suppose $\angle BAC$ is greater than $\angle BAD$.

Let $E$ be constructed on $AB$ such that $\angle BAE$, in the plane through $BA$ and $AC$, is equal to $\angle DAB$.

Let $AE = AD$.

Let $BEC$ be drawn across through $E$ such that the straight lines $AB$ and $AC$ are cut at $B$ and $C$.

Let $DB$ and $DC$ be joined.

We have that:

$DA = AE$

and $AB$ is common.

So we have two sides equal to two sides, while $\angle DAB = \angle DAE$.

$DB = BE$
$BD + DC > BC$

But we have that $BD = BE$.

Therefore $DC > EC$.

We have that:

$DA = AE$

and $AC$ is common.

We also have that $DC > DE$.

Therefore from Proposition $25$ of Book $\text{I}$: Converse Hinge Theorem:

$\angle DAC > \angle EAC$

But:

$\angle DAB = \angle BAE$

and so:

$\angle DAB + \angle DAC > \angle BAC$

Similarly it can be shown that the remaining plane angles, taken two together, are greater than the remaining one.

$\blacksquare$

## Historical Note

This theorem is Proposition $20$ of Book $\text{XI}$ of Euclid's The Elements.


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# Fault Tolerance¶

Currently Mars supports two levels of fault tolerance: process-level and worker-level. Scheduler-level support is not implemented now.

## Process-level Fault Tolerance¶

Mars uses multiple processes in its worker. When a worker process fails, for instance, gets killed because of out of memory, and the process is not Process 0 where control actors run, Mars worker will mark relevant tasks as failed, start another process, restore actors on it and the retry mechanism will restart the failed task.

## Worker-level Fault Tolerance¶

Note

New in version 0.2.0a2

As Mars uses execution graphs to schedule tasks, when some workers fail, scheduler will find lost chunks and work out affected operands. After that the spotted operands are rescheduled.

When a worker fails to respond, actors both in other workers and schedulers will detect it and send a feedback to ResourceActor, where changes in worker list is accepted and broadcast into all the sessions. When SessionActors accept the message, they will collect keys of missing data and relay all collected information to running GraphActors, where fail-over decisions are actually made.

The failure notification procedure is illustrated in the graph below.

When accepting a fail-over call, a GraphActor will first try reassigning states of affected operands, reassigning initial workers for initial operands, and then send updates to corresponding OperandActors to rerun these operands.

### Reassigning States¶

When some workers fail, data stored in these workers are lost. Therefore we need to change the states of these operands in order to run them again. As is stated in operand states, data generated by an operand only exist when operands are in FINISHED state, we perform a two-pass scanning procedure to calculate new assignments for operands:

1. Scan all FINISHED operands whose data are lost and mark them and their successors as affected;
2. Scan the graph from bottom to top, starting from affected operands;
3. For every affected operand, we scan its predecessors. If the data of the predecessor is lost or in a state without data or generating data, for instance, FREED or UNSCHEDULED, the predecessor will be marked as affected;
4. Mark current operand as READY when no predecessors are affected, otherwise it will be marked as UNSCHEDULED;
5. When there are no affected operands to be scanned, stop, otherwise go to Step 2.

### Reassigning Initial Workers¶

When workers fail, some of initial operands assigned in graph preparation step may not have workers to execute on. What’s more, change in number of workers may lead to imbalance in worker assignments. We solve this problem by applying an adaptive worker reassigning strategy. The algorithm framework is similar with the original one except that we do not visit initial operands which are executed, and the stop criterion of visiting operands in Step 3 is now limited to the average number of operands per worker minus the number of operands already executed or determined to run in the candidate worker.


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## Discriminants, resultants, and multidimensional determinants Gelfand, I. M. ; Kapranov, M. M. ; Zelevinsky , A. V. | Birkhäuser 1994

Ouvrage

V

- 523 p.
ISBN 978-0-8176-3660-9

Mathematics

Localisation : Ouvrage RdC (GELF)

A-discriminant # A-résultant # déterminant multidimensionnel # hyperdéterminant # polytope # variété de Chow

#### Filtrer

##### Codes MSC

Ressources Electroniques

Books & Print journals

Recherche avancée

0
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## Class field theory :from theory to practice Gras, Georges | Springer 2003

Ouvrage

V

- 491 p.
ISBN 978-3-540-44133-5

Springer monographs in mathematics

Localisation : Ouvrage RdC (GRAS)

théorie du corps de classe # idèle # corps de classe de rayon # morphisme d'Artin # symbole de Hasse # symbole de norme résiduelle # théorie des genres

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## Collected papers of Taro Morishima Morishima, Taro ; Karomatsu, Y. | Queen'S University 1990

Ouvrage

V

- 227 p.

Queen's papers in pure and applied mathematics , 0084

Localisation : Oeuvres complètes RdC (MORI)

classe d'ideaux # conjecture de Fermat # corps de classe # nombre de classe # théorie des corps cyclotomiques # théorème de Fermat # Morishima # oeuvres complètes

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## Computational algebraic theory Pohst, Michael E. | Birkhäuser 1993

Ouvrage

V

- 88 p.
ISBN 978-3-7643-2913-6

DMV Seminar , 0021

Localisation : Séminaire RdC

théorie algébrique des nombres # polynôme # géométrie des nombres # calcul en groupe d'unité # calcul en groupe de classe # factorisation de Pollard

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## Die berechnung der klassenzahl abelscher körper über quadratischen zahlkörpern Meyer, Curt | Akademie Verlag 1957

Ouvrage

V

- 132 p.

Mathematische lehrbücher und monographien , 0005

Localisation : Ouvrage RdC (MEYE)

calcul du nombre de classe # classe d'anneau # classe de rayon # corps abélien # corps de nombre quadratique # fonction l # forme de nombre de classe analytique # forme limite de Kronecker # loi de formation arithmétique et comportement analytique des # sommation des séries l # théorie algébrique des nombres # théorie des nombres # type de corps et structure

s" style="min-width:200px;">

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|

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## Stark's conjectures :recent work and new directions#an international conference on... held at Johns Hopkins University#Aug. 5-9 Burns, David ; Popescu, Cristian ; Sands, Jonathan ; Solomon, David | American Mathematical Society 2004

Congrès

V

- 221 p.
ISBN 978-0-8218-3480-0

Contemporary mathematics , 0358

Localisation : Collection 1er étage

théorie des nombres # conjecture de Stark # L-fonction # théorie d'Isawara

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## Class number statistics for imaginary quadratic fields Kurlberg, Pär | CIRM H

Multi angle

y

Research talks

The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine Soundararajan’s conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number $F(G)$ of imaginary quadratic fields with class group isomorphic to a given finite abelian group $G$.
Using Watkins’ tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance $(\mathbb{Z}/3\mathbb{Z})^3$ does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine the Cohen-Lenstra heuristics with a refinement of Soundararajan’s conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for families of $p$-groups as $p$ tends to infinity. For instance, it appears that no groups of the form $(\mathbb{Z}/p\mathbb{Z})^3$ and $p$ prime occurs as a class group of a quadratic imaginary field.
Conditionally on the Generalized Riemann Hypothesis, we extend Watkins’ data, tabulating $F(h)$ for odd $h \le 10^6$ and $F(G)$ for $G$ a $p$-group of odd order with $|G| \le 10^6$. (To do this, we examine the class numbers of all negative prime fundamental discriminants $-q$, for $q \le 1.1881 \cdot 10^{15}.$) The numerical evidence matches quite well with our conjectures.
This is joint work with S. Holmin, N. Jones, C. McLeman, and K. Petersen.
The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine So...

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## Congruent number problem and BSD conjecture Zhang, Shou-Wu | CIRM H

Multi angle

y

Research talks

A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang. A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has ...


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# Image of an involutive subbundle is involutive

Let $$f:M\to N$$ be a surjective submersion smooth map between two smooth manifolds $$M$$ and $$N$$. Consider the differential map $$df:TM\to TN$$ between the tangent bundles. Given an involutive subbundle $$A$$ of $$TM$$. Is $$df(A)$$ an involutive subbundle of $$TN$$?

There is no reason why $$df(A)$$ should even be a subbundle. For instance, let $$f:\mathbb{R}^{2}\rightarrow\mathbb{R}:(x,y)\mapsto x$$ be the projection and take $$A=\text{Span}(\partial_{y}+x\partial_{x})\subset T\mathbb{R}^{2}$$. Then $$df(A)$$ is zero at $$0\in\mathbb{R}$$, but one-dimensional at other points $$p\in\mathbb{R}$$.


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# What does this bandpass signal equation indicates?

This is the equation for bandpass signal from transmitter. What are the values for different parameters present in this equation. For example, what should be the value of P, Sl(t), Fc etc. Also what is the meaning of j in this equation.

Source for context.

• Don't think the answer is already present in the image. Example $f_{c}$ is center frequency – Mahendra Gunawardena Sep 11 '19 at 10:43
• Thanks for reply@ Mahendra , but what is the value for average transmitted power and Sl(t). – charu Sep 11 '19 at 11:44
• @mahendra I think Sl(t) will also have some equation. – charu Sep 11 '19 at 11:45
• @charu, I suggest the you use LaTex format to improve readability for symbols such as $\rho$ – user8055 Sep 11 '19 at 13:36
• @user8055 sure ....from next time I will incorporate this. – charu Sep 12 '19 at 14:32


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# Kooperativer Bibliotheksverbund

## Berlin Brandenburg

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Description: According to experimental observations, capillary trapping is strongly dependent on the roughness of the pore–solid interface. We performed imbibition experiments in the range of capillary numbers ( Ca ) from $$10^{-6}$$ 10 - 6 to $$5\times 10^{-5}$$ 5 × 10 - 5 using 2D-micromodels, which exhibit a rough surface. The microstructure comprises a double-porosity structure with pronounced macropores. The dynamics of precursor thin-film flow and its importance for capillary trapping are studied. The experimental data for thin-film flow advancement show a square-root time dependence. Based on the experimental data, we conducted inverse modeling to investigate the influence of surface roughness on the dynamic contact angle of precursor thin-film flow. Our experimental results show that trapped gas saturation decreases logarithmically with an increasing capillary number. Cluster analysis shows that the morphology and number of trapped clusters change with capillary number. We demonstrate that capillary trapping shows significant differences for vertical flow and horizontal flow. We found that our experimental results agree with theoretical results of percolation theory for $$Ca =10^{-6}$$ C a = 10 - 6 : (i) a universal power-like cluster size distribution, (ii) the linear surface–volume relationship of trapped clusters, and (iii) the existence of the cutoff correlation length for the maximal cluster height. The good agreement is a strong argument that the experimental cluster size distribution is caused by a percolation-like trapping process (ordinary percolation). For the first time, it is demonstrated experimentally that the transition zone model proposed by Wilkinson (Phys Rev A 30:520–531, 1984) can be applied to 2D-micromodels, if bicontinuity is generalized such that it holds for the thin-film water phase and the bulk gas phase.


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# joint, marginal, and conditional relative frequencies worksheet pdf

477 Excel The marginal relative frequency is expressed as both a decimal and a percentage value . Two types of Relative Frequencies are found in a relative frequency table: A Marginal Relative Frequency is found by dividing a row total or a column total by the grand total. . *Click on Open button to open and print to worksheet. A conditional relative frequency is the ratio of a joint relative frequency to the marginal relative frequency. Of Say there are 20 people among adults and children and each was given a choice to either eat chicken or beef for the ball. Probabilities, such as these simple and joint probabilities, have no store, and that the sample size was very large (much larger than In the table, the inner cells represent the joint relative frequency. D the long term relative frequency into a percent. Complete the table below. Example: Given the student is in the 11th grade, what is the probability they like math? Categorical Variables The next subject is inferential statistics. joint, marginal, and conditional relative frequencies worksheet pdf. So, about 63.33% of the participants were men, and from the total population, about 36.67% struggled with math the most in their high school. The Joint probability is a statistical measure that is used to calculate the probability of two events occurring together at the same time P(A and B) or P(A,B). To calculate the joint relative frequency, take the ratio between the joint frequency to the total frequency of the data. Worksheets are Joint relative marginal relative conditional relative, Accelerated algebrageometry a two way frequency tables, Joint conditional marginal probabilities, Two way tables and probability, Two way tables walking and bicycling to work teacher, , Two way frequency tables work name tim busken, 1 exploration . frequencies and that the events within a categorical variable are making inferences. frequency relative worksheet tables worksheets quiz study table plot dot practice definition examples chart identify fedjp grass site. She asks her friends, 12 boys and eight girls, which geometry teacher they would . Example: Of the students that like English, what percent are in the 10th grade? C B. You remember, confidence Levine, D., Berenson, M. & Stephan, When attempting to compare all three types of relative frequencies: Joint, Marginal, and Conditional, one must consider the size of the sample and how the data is spread out. P. 360 A Joint Relative Frequency is found by dividing a frequency that is not in the Total row or the Total column by the grand total. Interpret relative frequencies in the context of the data . 3 boys selected clown. arrangements when there are two or more groups. PDF. What is the relative frequency of students that do not do chores nor get an allowance to the total number of students? The joint probabilities occurs in the body of the cross-classification table at the intersection of two events for each categorical variable. Cincinnati, OH: South-Western, Chapter 4 and Chapter 5. Correct! Relative frequency is normally represented as a decimal or a percent. generally accepted practice by writing the probability of Excellent As a member, you'll also get unlimited access to over 84,000 891 All of The dataset in consideration here is represented in the form of a two way table. Roberto has worked for 10 years as an educator: six of them teaching 5th grade Math to Precalculus in Puerto Rico and four of them in Arizona as a Middle School teacher. Conditional relative frequency numbers are the ratio of a joint relative frequency and related marginal relative frequency. Oct 23 2017 I created this worksheet to shield my students practice organizing. Quiz & Worksheet - Joint, Marginal & Conditional Frequencies | Study.com study.com. The conditional relative frequency is 4 / 6 = .67 or 67%. Great! 1 0 obj 15A 15B 15C 15D 15E Relative frequency and probability. The E 207 457 For example, for the fixed level Z = k, the conditional odds ratio between X and Y is. Assumptions 0.15. 0.10. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). This is a wk:rNY:K iVFf_](McC) U}50}Nd4 The combinations are Kmart/Sears; Kmart/JCP; Analysts use conditional relative frequency to determine if there is a trend involving two distinct variables of categorical data. 19. E (Random) Frequency Tables a. Construct a row conditional relative frequency table of the above data. To do this, he needs to understand joint, marginal, and conditional frequencies. Joint Probability Use this scavenger hunt as a way to get students out of their seat and moving around the classroom! Sears/Kmart; Kmart/JCP; JCP/Kmart; Kmart/Wards; Wards/Kmart; Joint relative frequency is the ratio of the frequency in a particular category and the total number of data values. Our last module for the course (did I hear loud variables? is the subject of Module Notes 5.2. Joint Marginal Conditional Frequencies Definitions Two Way Tables S Id 5 High School Math Teachers See also Maths Times Tables Grid. data were measured on an interval scale. When you think of marginal frequency, think of the margins on an English paper. I would definitely recommend Study.com to my colleagues. within a categorical variable are presented. joint probabilities. The tools we introduce in To find the first conditional relative frequency, it asks what part of people who struggle in science are men. /SMask /None>> 477 This is a similar set up to conditional probability, where the limitation, or condition, is preceded by the word given. 5. that shoppers give an Excellent rating, when we consider all of the Remember, the joint relative frequency numbers are in the purple cells, and the marginal relative frequency numbers are in the green cells. Good 205 Joint relative frequency is the ratio of the frequency in a particular category and the total number of data values. This product contains 15 problems on relative frequency (joint relative, marginal relative, and conditional relative are all included). Conditional probability is the probability of one event occurring in the presence of a second event. There are three types of relative frequency: The best statistical instrument available when collecting, comparing, and analyzing pairs of qualitative data is a two-way table. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment. b) Find the relative frequency. 3 lessons in math, English, science, history, and more. The 18 Enrolling in a course lets you earn progress by passing quizzes and exams. different ways can shoppers visit just two of the four stores if number of data values. The last probability can occur whenever we are using comparing the count data 1301 to 1808 to 891. Oct 2111:29 AM Girls Boys Total observations for all possible events in the sample space. The data could be marks scored by students, temperatures of different towns, points scored in a volleyball match, etc. /ColorSpace /DeviceRGB First, look back at our frequency table, not the relative frequency table. 5 Question 1. f. Use the table to find the marginal relative frequency of students surveyed who prefer vacationing at the seashore. Worksheet, I enter =C3/F7 in cell C14. JCP Techniques in Business and Economics (10th. Conditional relative frequency is found by calculating a division between a joint relative frequency from the inner cells and either a row or a column total. For example, let's say you wanted to find the percentage of people that selected clown as a career, given those people are girls. 5 Note that I added marginal totals to the For joint relative frequency, divide one cell of category pairs bythe total frequency. Other material covered in reference texts includes probability Conditional Odds Ratios. For example, the condition of interest in the first row is females. The only assumptions that we need for computing these The marginal relative frequency of a data set is calculated by dividing the sum or total of a row or the sum or total of a column by the total number of observations in a dataset. 391 /CA 1.0 Joint Probability. Joint, Marginal, and Conditional Probabilities - YouTube 11.2 experimental probability #46, Marginal, Joint and Conditional Probability | BUSINESS STATISTICS . Displaying all worksheets related to - Marginal And Joint Frequencies. BThey describes the relationship between the two categorical variables. In this lesson you will learn how to use two-way frequency tables by determining if the data is a frequency count or a relative frequency. It is called a marginal What is the probability that a Good Good We can convert 0.325 into a percent by multiplying by 100. marginal worksheet joint frequencies conditional polymorphism quiz practice java inheritance vs study definitions differences . This is called the conditional probability. Travis wants to know the joint relative frequency of the boys that selected elephant tamer. The two-way table below shows the number of students with each hair color and eye color. This book is intended to both serve as a reference guide and a text for a course on Applied Mathematical Programming for upper undergraduate and Sears 25.1% Recall that events have to be mutually exclusive, so if a shopper Answer: ___________________________________________________________________________________________________________________________________________. A permutation is any arrangement of r objects Causation in Statistics: Overview & Examples | What is Causation? average them, or find their standard deviation, or their median - 2. 47.7% The methods for solving . So, we aren't looking at the total number of people that responded to the survey, only the girls in this case. Simple Probability Create a two-way relative frequency table that summarizes the data from the table. In this conditional relative frequency, we are basically asking the question, 'What percentage of people that selected clown are boys?'. Worksheet 5.1.5 Travis also wants to know the number of people surveyed that were boys, given those people chose clown as a career. Use the above table to find each conditional relative frequency. 13 \$3.00. the analysis of categorical (also called qualitative) data. 6 rules from mathematics and statistics (Mason, 1999). CONDITIONAL RELATIVE FREQUENCY WORKSHEET. conditional probabilities. The horizontal categorical variable shows the careers and the vertical categorical variable shows the genders. shoppers, is 1301 divided by 4000, or 0.325 gave an Excellent rating. 39.1%. Now, can you identify the marginal relative frequency of the people that selected acrobat as a career? Recognize possible associations and trends in the data. Most Popular Cartier Jewelry, Worksheet will open in a new window. Excel Circumscribed Angle Theorem & Calculation | What is a Circumscribed Angle? win or lose on the next spin) (or they enjoy the free food and space of interest. 14 In the table above, the marginal frequencies can be found using the bottom or far-right "total" columns. However, you can also use them to calculate joint, marginal, and conditional probabilities! Table 1. 100% Total We simply count their . When analyzing data in a two-way frequency table, you will be looking for joint relative frequency, which is the ratio of the frequency in a particular category and the total number of data values. 40.4% Printable Exercises. JCP 457 Create a two-way relative frequency table that summarizes the data from the table. shoppers who rated Sears as Good. So, ed.). categorical variable - its values are categories or names - we can't The joint relative frequency is .40 or 40%. A Conditional Relative Frequency can be found by dividing the joint relative frequency by the marginal relative frequency. 100%. In statistics, this is used to compare and pair two distinct types of categorical data. 2.) Row 12 Total Right! | 9 2 /SM 0.02 F (Random) Methods of Data Collection a. Census b. grand total . They are used to determine associations between categorical data. Problem 1 : A survey is conducted among school students. We will cover one of these, To compute the marginal probability in Cell C13, using the But there are simple descriptive statistics for categorical variables CThey organizes data into classes and shows the frequency of each class. That To get the relative frequency, divide each frequency by the total. multiple regression model. Excellent by the total shoppers at Wards which gives 391/1000 or formula applies to arrangements when there is only one group. SPORTS. Excel. The joint pdf is, similarly, fX,Y(x,y) = P(X = x and Y = y). 228 bishop in budget plan worksheet two way tables and relative frequency worksheet answersdownload by size handphone, a two dice are rolled and a total of 14 is obtained relative frequency of ne weather on christmas day think write 1 write the formula relative frequency calculate the relative frequency of a having mechanical problems in the fi rst . Now, find the number of girls that selected elephant tamer. Marginal relative frequency is the ratio of the sum of the joint relative frequency in a row or column and the total number of data values. 8th Grade Math Relative Frequency Practice 1. Excellent Rating to 45.2% for Good to 22.3% for poor than by Let's not get to technical for this simple concept. The table given below shows the results of the survey. On the other hand, the MCLE is consistent for the conditional distribution P(YjX) alone, and pro-vides no information about either the joint or the marginal distributions. >> Proportion, 5.3: 100.0% 26 Excellent, 2 = Good, and 1 = Poor. 20.9% zero + 1 double zero) = 18/38 = 0.474 or 47.4%. This worksheet will assist you in better understanding how to determine the joint, marginal and conditional relative frequencies. Worksheets are Joint relative marginal relative conditional relative, Accelerated algebrageometry a two way frequency tables, Joint conditional marginal probabilities, Two way tables and probability, Two way tables walking and bicycling to work teacher, , Two way frequency tables work name tim busken, 1 exploration . The multiplication rule applies to finding the number of /Creator ( w k h t m l t o p d f 0 . The Rating variable has three "values": Excellent, Good and 25.0% 22.3% /Width 625 In simpler words, it explains how many members of a marginal frequency have a particular trait. Therefore, we must use the total in the acrobat column to create our conditional relative frequency. A graphical display that shows the distribution of a categorical variable by displaying the frequency or relative frequency of each category as a bar. when we do so, we have to recognize that the numbers assigned are Good joint marginal and conditional relative frequencies, in this lesson we learn about categorical data and how it can be organized in two way frequency tables relative frequency terminology including join marginal and conditional is taught in context, unit 13 two way tables faculty guide page 3 the video solutions 1 Now I know that more shoppers rated the four exhaustive assumptions, all probabilities for all events within the Note there are some other percents or Learn the joint, marginal, and conditional relative frequency definitions and see how to identify each in a two-way table. An identification activity will be given to the learners to determine the relative frequencies from the given definition. We wrap up this module by studying descriptive and You would then find the number of girls that selected clown and divide by the total number of girls in the survey. Mathematical expectation of functions of random vectors. as P(Excellent). of Module 3 - recall we incorporated a "dummy" variable to represent If a gambler bets that a JCP /Subtype /Image For example, we can see: There were a total of 13 respondents who were male and preferred baseball. Joint moments, Covariance and Correlation. Lesson 11 Conditional Relative Frequencies November 19, 2013 Let students work in pairs. The purple cells on this table are all joint frequency numbers. Relative Frequency, Joint Frequency, Marginal Frequency, and Conditional Frequency Use the following data for # I - 21, 17, 3, 26 21, 32 I. Hace un ao. Same is true for the For courses in Business Statistics. The numbers in the column on the very right and on the row on the very bottom are the marginal frequency numbers. Related post: Detecting Relationships in a Contingency Table What is the "Joint, Marginal, And Conditional Relative Frequencies Worksheet (With Answer Key pdf)"? Total Upper Saddle River, NJ: Prentice-Hall, Chapter 4. Use two way frequency tables frequency count vs . This is a similar set-up to conditional probability, where the limitation, or condition, is preceded by the word given. Col B and we cover them in this module. Cluster Summarize, represent, and interpret data on two categorical and quantitative variables. To compute this probability in cell C14 of the The full text downloaded to your computer With eBooks you can: search for key concepts, words and phrases make highlights and notes as you study share your notes with friends eBooks are downloaded to your computer and accessible either offline through the Bookshelf (available . This is called joint frequency because you are joining one variable from the row and one variable from the column. Interpret. /CreationDate (D:20210408072513+03'00') The joint prob. 5.1.3 is a copy of Worksheet 5.1.2 in rows 12 to 18 of the same Excel You can & download or print using the browser document reader options. PERCENT OF TOTAL Printable Exercises. . Suitable for Algebra 1 students, grade 8th and grade 9th.This m You may be asking the question, 'How do you know which total to pick as the denominator, the column or the row?' frequency table which displays conditional frequencies for gender. Showing top 8 worksheets in the category - Marginal And Joint Frequencies. The joint frequencies are the frequencies shown in the cells of the table: These values are known as "joint" frequencies because they tell us the frequency of two values that occur jointly. Illustrative Mathematics. 32.3% The permutation Best Places To Eat In Lancaster, Pa, Kmart Summarize categorical data for two categories in two-way frequency tables. D 8 . Poor. 25% 228 D )Y"""dtvSy . Create a relative frequency table. Question 1 : A survey is conducted among school students. Side note: I don't think people who gamble believe in long term Click on pop-out icon or print icon to worksheet to print or download. occurrences in a way that provides useful information. Frequencies and Relative Frequencies in Histograms. Exercises 11-16 11. Because of the mutually exclusive and About Worksheet Probability Pdf Conditional Kuta . different ways can shoppers visit the four stores if order 207 applications of categorical variables. However, for this module, we are not going to assume we can convert Another example: what is P(Good or 4 Specifically, you learned: Joint probability is the probability of two events occurring simultaneously. two categorical variables. 100.0% C 9th Grade 10th Grade 11th Grade 12th Grade TOTAL Excellent, you cannot also rate it as Good - an observation has to /Filter /DCTDecode categorical event space must sum to 100%. we will look at that later). Disadvantages Of Pressure Transducer, All other trademarks and copyrights are the property of their respective owners. oT nd the relative frequencies, divide each frequency by the total number of students in the sample - in this case, 20. Two Way Frequency Table Worksheet. The application is in decision trees. Confidence Interval and Hypothesis Testing for a What is the conditional relative frequency of females with an "Excellent" social life rating? Joint Marginal And Conditional Relative Frequencies Worksheet (With Answer Key pdf), Fraction Worksheet For Class 7 (With Answer Key), Unit 6 using the unit circle worksheet 7 (with answer key), 2nd Grade Measuring Worksheet (with Answer Key), Square Numbers Worksheet (with Answer Key), Expanded Form Worksheet (with Answer Key). Search: Relative Frequency Histogram Worksheet. Poor Good. In this example, the row variables are the careers and the column variables are the genders. . A marginal relative frequency is the sum of the joint relative frequencies in a row or a column. Worksheets for Algebra I, Module 2, Lesson 11 (pdf) Student Outcomes. Rather, we are talking about probabilities Find the joint frequencies of the women who love soda? just divide the number of shoppers who rated the Wards experience as follow to visit two of four stores, if order is not to ask, what is the probability that a shopper rates their experience Two Way Frequency Tables December 08, 2014 Two-Way Table Definitions: Relative Frequency- a ratio comparing one thing to another (usually written as a decimal) Joint Relative Frequency- The ratio of the frequency in a particular category and the total number of data values (ex: what percent of the students have blue eyes and blond Recognize possible associations and trends in the data. order matters. Travis uses a two-way table to analyze his data. E survey 1000 shoppers at four stores in Fort Myers several years ago. assign a value to this type of categorical variable, such as 3 = here. 25.3% Marginal and Joint Frequencies for two-way Tables c. Conditional Relative Frequencies and Association (and Simpson's Paradox) d. Comparing Distributions Using Bar Charts. I compute descriptive statistics such as the average rating. permutations. 4. 3 0 obj We use a simple example to explain conditional probabilities. 9.8% Total S-ID. 32.5% To calculate the joint. 25% We can also convert a count 205 404 four stores given the rating. - Definition & Examples, Joint, Marginal & Conditional Frequencies: Definitions, Differences & Examples, McDougal Littell Pre-Algebra: Online Textbook Help, Common Core Math Grade 8 - Expressions & Equations: Standards, Common Core Math Grade 8 - Functions: Standards, CSET Math Subtest III (213): Practice & Study Guide, CLEP College Algebra: Study Guide & Test Prep, CLEP College Mathematics: Study Guide & Test Prep, UExcel Statistics: Study Guide & Test Prep, CAHSEE Math Exam: Test Prep & Study Guide, Marginal & Conditional Probability Distributions: Definition & Examples, Undetermined Coefficients: Method & Examples, Oscillation: Definition, Theory & Equation, Strategies for Reading Comprehension Passages on the LSAT, Strategies for Analytical Reasoning Questions on the LSAT, Recognizing When Two Statements Are Logically Equivalent, Strategies for Logical Reasoning Questions on the LSAT, Formal Logic Problem Solution: Steps & Tips, Recognizing Misunderstandings & Points of Disagreement, Calculating the Square Root of 27: How-To & Steps, Working Scholars Bringing Tuition-Free College to the Community, Convert your numbers into relative frequency numbers, Calculate conditional relative frequencies, Remember the steps to follow when using a two-way table to analyze data, Create two-way frequency and two-way relative frequency tables, Calculate the joint, marginal and conditional relative frequencies of a data set. For example, a consumer products company was hired to There are also inferential P. 360 A Joint Relative Frequency is found by dividing a frequency that is not in the Total row or the Total column by the grand total. the events good and poor are marginal events for the same category. 5) 4 323 Using Two-Way Tables to Evaluate Independence, Practice Making Geometric Constructions with Tools. a) What is the joint relative frequency of girls who have a cat? 5.1: Simple, Joint, Marginal and F represents the word "given" which provides the subset of the event Wards Bill Russell Auctioning Memorabilia, It also helps with analysis if there is an association regarding the kinds of variables available in the data collection, and it is useful for easily finding ratios to compare different pairs of relative frequencies. 6 0 obj Sears F Marginal relative frequency is the ratio of the sum of the joint relative frequency in a row or column and the total number of data values. Create your account. cross-classification table at the intersection of two events for each The conditional statement will give you the clue. This should be equivalent to the joint probability of a red and four 252 or 126 divided by the marginal P red 12. These are usually larger than joint frequencies, since they both are compared with the grand total of the data. relative frequencies. generally get more relative information by comparing 32.5% for Worksheet 5.1.2 provides some more ways of counting the survey Two way relative frequency tables practice Khan Academy. On this table, the marginal frequency numbers are in the green cells. The long term chance Joint Relative Frequency is the frequency of the intersection between two types of categorical data. Marginal And Joint Frequencies - Displaying top 8 worksheets found for this concept. A 1 A 2 Total B 1 a/n b/n (a+b)/n B 2 c/n d/n (c+d)/n Total (a+c)/n (b+d)/n 1 The marginal probability of A 1. Using the data in the table above, find the relative frequency of (a) men, given that they struggle in science, and (b) people who struggle in math, given that the person is a woman. Students calculate and interpret conditional relative frequencies from two-way frequency tables. Kmart tomorrow and finding the experience excellent, since the For example, you can see that out of all of the people surveyed, 9 chose elephant tamer as a career. PERCENT OF ROW TOTALS: 323 22.8% The number of males in the survey. The table given below shows the results of the survey. Worksheets are Joint relative marginal relative conditional relative, Accelerated algebrageometry a two way frequency tables, Joint conditional marginal probabilities, Two way tables and probability, Two way tables walking and bicycling to work teacher, , Two way frequency tables work name tim busken, 1 exploration completing and using a two way table. Row 1 $$(a)\;Marginal\;Relative\;Frequency=\frac{Men}{Total\;Frequency} = \frac{95}{150}\approx 0.6333 \\ (b)\;Marginal\;Relative\;Frequency=\frac{Math}{Total\;Frequency} = \frac{55}{150}\approx 0.3667$$. Problem 1 : A survey is conducted among school students. probability in cell C25, I enter =C3/C7 in cell C25. Sears, JCP, Wards, then Kmart. Two Way Table amp Relative Frequency HW Daily Homework. A graphical display that shows the relationship between two categorical variables by dividing the area of a rectangle into tiles that represent the different categories of both variables. _________________________________________________________________________. Worksheet 5.1.3 Find the joint relative frequencies of (a): women who struggled in English in high school and (b) men who struggled in science in high school. Recognize possible associations and trends in the data. The Permutation Formula You can find a conditional relative frequency using a row total or a column total of a two-way table. Total Relative Frequency Scavenger Hunt (Joint, Marginal, & Conditional) by. variables, how do we measure them. [/Pattern /DeviceRGB] You are able to edit this assignment using Google Forms and assign it to your students via Google Classroom.These two-way frequency table task cards ask students to find:Joint, marginal, and conditional relative frequenciesPercentages using the tables12 task cards + answer keysYou may also like:Two-Way Frequency Table Notes & WorksheetsBox and Whi A conditional relative frequencycompares a frequency count to the marginal total that represents the condition of interest. david brooks health 2021, johnson memorial hospital enfield, ct, Saddle River, NJ: Prentice-Hall, Chapter 4 and Chapter 5 are marginal events for each categorical shows! Count 205 404 four stores in Fort Myers several years ago Way table amp relative of. 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And a percentage value other trademarks and copyrights are the marginal joint, marginal, and conditional relative frequencies worksheet pdf red 12 zero =. Several years ago usually larger than joint frequencies eye color, practice making Geometric Constructions with.... Use a simple example to explain conditional probabilities - YouTube 11.2 experimental #. D the long term chance joint relative frequency table that summarizes the data 5 High school math See! Using the bottom or far-right  total '' columns 1808 to 891 worksheet. Statistics ( Mason, 1999 ) exclusive and About worksheet probability pdf Kuta! About worksheet probability pdf conditional Kuta 15B 15C 15D 15E relative frequency can be found using the bottom far-right. 1301 to 1808 to 891 marginal & amp ; conditional ) by ( Random ) Methods of data a.., worksheet will open in a volleyball match, etc 3 0 obj 15A 15C...  total '' columns of Pressure Transducer, all other trademarks and are... 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# American Institute of Mathematical Sciences

March  2006, 1(1): 57-84. doi: 10.3934/nhm.2006.1.57

## Numerical approximations of a traffic flow model on networks

 1 Department of Engineering of Information and Applied Mathematics, DIIMA, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy 2 Istituto per le Applicazioni del Calcolo "M. Picone", IAC-CNR, Viale del Policlinico, 137, 00161, Roma, Italy 3 Istituto per le Applicazioni del Calcolo "M. Picone", IAC-CNR, Viale del Policlinico 137, 00161 Roma

Received  September 2005 Revised  October 2005 Published  January 2006

We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occur and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.
Citation: Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57
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text:
[–] -1 points0 points ago

sorry, this has been archived and can no longer be voted on

multiply both sides by sin(x)(1+cosx)

(1-cosx)(1+cosx) = (sinx)2

(1-cosx)(1+cosx) = 1 - (cosx)2

get the (cosx)2 to the other side

1 = (sinx)2 + (cosx)2 and thats a true identity.

[–] 0 points1 point ago

sorry, this has been archived and can no longer be voted on

Operating on both sides like this (and moving terms from one side to the other, as you suggest) assumes that the identity is true prior to verifying it.

Best practices typically require working only on one side of the identity.

[–] 1 point2 points ago

sorry, this has been archived and can no longer be voted on

I sincerely disagree with that, here comes a big rant (not supposed to be offensive), so who doesn't like rants, feel free to ignore it.

"Operating on both sides like this" doesn't assume anything. Using equality relation when we don't know the truth of such equality isn't misuse of notation, in mathematics we aren't restricted to true logical statements, there often appear statements in mathematical arguments of which we are not sure about their truth value (or they may be simply plainly wrong!), but after we are done with forming the argument, it will be clear what is true and what not (if we succeed), it simplifies many arguments and makes them more clear. Also note that operating on both sides of true equality relation and not-equal-to relation is the same!

Operating on both sides is valid because the equations are equivalent (on certain domain of interest). Let me illustrate what I mean. Here are logical connectives between the equations (equations of user allnicksaretaken) if we assume that [; x ;] is any real number such that the identity we wish to derive is well defined (we obviously aren't interested in others), then:

[; \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x} \Leftrightarrow (1 - \cos x)(1 + \cos x) = \sin^2 x \Leftrightarrow (1 - \cos x)(1 + \cos x) = 1 - \cos^2 x \Leftrightarrow 1 = \sin^2 x + \cos^2 x ;]

See? There are only equivalences, we didn't assume anything. We know that this compound statement is true by laws of elementary algebra and we also know that last statement in the chain is true, so the first equality follows.

This identity isn't best example, because working with both sides doesn't really simplify the proof, but in more tricky cases if we work with both sides of equality, then it often simplifies the proofs.

EDIT: The equivalences (along with quantification) are usually omitted, because people are lazy and they are pretty much obvious usually.

[–] 0 points1 point ago

sorry, this has been archived and can no longer be voted on

Technically true, but as long as you never did anything like multiply both sides by 0, you can just reverse the order of the lines and get a valid proof.

[–] 10 points11 points ago

sorry, this has been archived and can no longer be voted on

Multiply the left side by (1+cosx)/(1+cosx)

[–] 3 points4 points ago

sorry, this has been archived and can no longer be voted on

Working with both sides of an "identity" is a no-no, haha. Here's one approach to the problem.
(1-cosx)/(sinx)
Working with the left side only, we begin by multiplying top and bottom by sinx.
[sinx(1-cosx)]/(sin2 x)
Rewrite the bottom from the identity sin2 x + cos2 x = 1
[sinx(1-cosx)]/(1 - cos2 x)
Factor the bottom
[sinx(1-cosx)]/[(1 - cosx)(1 + cosx)]
Cancel out like terms,
sinx/(1 + cosx) = RHS
Tah-dah!