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Transportation engineering

Transportation engineering or transport engineering is the application of technology and scientific principles to the planning, functional design, operation and management of facilities for any mode of transportation in order to provide for the safe, efficient, rapid, comfortable, convenient, economical, and environmentally compatible movement of people and goods transport.

Theory
The planning aspects of transportation engineering relate to elements of urban planning, and involve technical forecasting decisions and political factors. Technical forecasting of passenger travel usually involves an urban transportation planning model, requiring the estimation of trip generation, trip distribution, mode choice, and route assignment. More sophisticated forecasting can include other aspects of traveler decisions, including auto ownership, trip chaining (the decision to link individual trips together in a tour) and the choice of residential or business location (known as land use forecasting). Passenger trips are the focus of transportation engineering because they often represent the peak of demand on any transportation system.

A review of descriptions of the scope of various committees indicates that while facility planning and design continue to be the core of the transportation engineering field, such areas as operations planning, logistics, network analysis, financing, and policy analysis are also important, particularly to those working in highway and urban transportation. The National Council of Examiners for Engineering and Surveying (NCEES) list online the safety protocols, geometric design requirements, and signal timing.

Transportation engineering, primarily involves planning, design, construction, maintenance, and operation of transportation facilities. The facilities support air, highway, railroad, pipeline, water, and even space transportation. The design aspects of transportation engineering include the sizing of transportation facilities (how many lanes or how much capacity the facility has), determining the materials and thickness used in pavement designing the geometry (vertical and horizontal alignment) of the roadway (or track).

Before any planning occurs an engineer must take what is known as an inventory of the area or, if it is appropriate, the previous system in place. This inventory or database must include information on population, land use, economic activity, transportation facilities and services, travel patterns and volumes, laws and ordinances, regional financial resources, and community values and expectations. These inventories help the engineer create business models to complete accurate forecasts of the future conditions of the system.

Operations and management involve traffic engineering, so that vehicles move smoothly on the road or track. Older techniques include signs, signals, markings, and tolling. Newer technologies involve intelligent transportation systems, including advanced traveler information systems (such as variable message signs), advanced traffic control systems (such as ramp meters), and vehicle infrastructure integration. Human factors are an aspect of transportation engineering, particularly concerning driver-vehicle interface and user interface of road signs, signals, and markings.

Highway engineering

Engineers in this specialization:
 * Handle the planning, design, construction, and operation of highways, roads, and other vehicular facilities as well as their related bicycle and pedestrian realms
 * Estimate the transportation needs of the public and then secure the funding for projects
 * Analyze locations of high traffic volumes and high collisions for safety and capacity
 * Use engineering principles to improve the transportation system
 * Utilize the three design controls, which are the drivers, the vehicles, and the roadways themselves

Railroad engineering
Railway engineers handle the design, construction, and operation of railroads and mass transit systems that use a fixed guideway (such as light rail or monorails).

Typical tasks include:

 * Determine horizontal and vertical alignment of the railways
 * Determine station location
 * Design functional segments of stations like lines, platforms, etc.
 * Estimate construction cost

Railway engineers work to build a cleaner and safer transportation network by reinvesting and revitalizing the rail system to meet future demands. In the United States, railway engineers work with elected officials in Washington, D.C., on rail transportation issues to make sure that the rail system meets the country's transportation needs.

Railroad engineers can also move into the specialized field of train dispatching which focuses on train movement control.

Port and harbor engineering
Port and harbor engineers handle the design, construction, and operation of ports, harbors, canals, and other maritime facilities.

Airport engineering
Airport engineers design and construct airports. Airport engineers must account for the impacts and demands of aircraft in their design of airport facilities. These engineers must use the analysis of predominant wind direction to determine runway orientation, determine the size of runway border and safety areas, different wing tip to wing tip clearances for all gates and must designate the clear zones in the entire port. The Civil Engineering Department, consisting of Civil and Structural Engineers, undertakes structural design of passenger, terminal design and cargo terminals, aircraft hangars (for parking commercial, private and government aircraft), runways and other pavements, technical buildings for installation of airport ground aids etc. for the airports in-house requirements and consultancy projects. They are even responsible for the master plan for airports they are authorized to work with.

title: Transportation engineering


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Queueing theory

Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Queueing theory has its origins in research by Agner Krarup Erlang, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of teletraffic engineering and have since seen applications in telecommunication, traffic engineering, computing, project management, and particularly industrial engineering, where they are applied in the design of factories, shops, offices, and hospitals.

Spelling
The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is Queueing Systems.

Description
Queueing theory is one of the major areas of study in the discipline of management science. Through management science, businesses are able to solve a variety of problems using different scientific and mathematical approaches. Queueing analysis is the probabilistic analysis of waiting lines, and thus the results, also referred to as the operating characteristics, are probabilistic rather than deterministic. The probability that n customers are in the queueing system, the average number of customers in the queueing system, the average number of customers in the waiting line, the average time spent by a customer in the total queuing system, the average time spent by a customer in the waiting line, and finally the probability that the server is busy or idle are all of the different operating characteristics that these queueing models compute. The overall goal of queueing analysis is to compute these characteristics for the current system and then test several alternatives that could lead to improvement. Computing the operating characteristics for the current system and comparing the values to the characteristics of the alternative systems allows managers to see the pros and cons of each potential option. These systems help in the final decision making process by showing ways to increase savings, reduce waiting time, improve efficiency, etc. The main queueing models that can be used are the single-server waiting line system and the multiple-server waiting line system, which are discussed further below. These models can be further differentiated depending on whether service times are constant or undefined, the queue length is finite, the calling population is finite, etc.

Single queueing nodes
A queue or queueing node can be thought of as nearly a black box. Jobs (also called customers or requests, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.

However, the queueing node is not quite a pure black box since some information is needed about the inside of the queueing node. The queue has one or more servers which can each be paired with an arriving job. When the job is completed and departs, that server will again be free to be paired with another arriving job.

An analogy often used is that of the cashier at a supermarket. (There are other models, but this one is commonly encountered in the literature.) Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting where a customer will leave immediately if the cashier is busy when the customer arrives, is referred to as a queue with no buffer (or no waiting area). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n.

Birth-death process
The behaviour of a single queue (also called a queueing node) can be described by a birth–death process, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If k denotes the number of jobs in the system (either being serviced or waiting if the queue has a buffer of waiting jobs), then an arrival increases k by 1 and a departure decreases k by 1.

The system transitions between values of k by "births" and "deaths", which occur at the arrival rates $$\lambda_i$$ and the departure rates $$\mu_i$$ for each job $$i$$. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single average rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate of $$\lambda = \text{avg}(\lambda_1,\lambda_2,\dots,\lambda_k)$$ and a departure rate of $$\mu = \text{avg}(\mu_1, \mu_2, \dots, \mu_k)$$.

Balance equations
The steady state equations for the birth-and-death process, known as the balance equations, are as follows. Here $$P_n$$ denotes the steady state probability to be in state n.

 * $$\mu_1 P_1 = \lambda_0 P_0$$
 * $$\lambda_0 P_0 + \mu_2 P_2 = (\lambda_1 + \mu_1) P_1$$
 * $$\lambda_{n-1} P_{n-1} + \mu_{n+1} P_{n+1} = (\lambda_n + \mu_n) P_n$$

The first two equations imply
 * $$P_1 = \frac{\lambda_0}{\mu_1} P_0$$

and
 * $$P_2 = \frac{\lambda_1}{\mu_2} P_1 + \frac{1}{\mu_2} (\mu_1 P_1 - \lambda_0 P_0) = \frac{\lambda_1}{\mu_2} P_1 = \frac{\lambda_1 \lambda_0}{\mu_2 \mu_1} P_0$$.

By mathematical induction,
 * $$P_n = \frac{\lambda_{n-1} \lambda_{n-2} \cdots \lambda_0}{\mu_n \mu_{n-1} \cdots \mu_1} P_0 = P_0 \prod_{i = 0}^{n-1} \frac{\lambda_i}{\mu_{i+1}}$$.

The condition $$\sum_{n = 0}^{\infty} P_n = P_0 + P_0 \sum_{n=1}^\infty \prod_{i=0}^{n-1} \frac{\lambda_i}{\mu_{i+1}} = 1$$ leads to
 * $$P_0 = \frac{1}{1 + \sum_{n=1}^{\infty}\prod_{i=0}^{n-1} \frac{\lambda_i}{\mu_{i+1}} }$$

which, together with the equation for $$P_n$$ $$(n\geq1)$$, fully describes the required steady state probabilities.

Kendall's notation
Single queueing nodes are usually described using Kendall's notation in the form A/S/c where A describes the distribution of durations between each arrival to the queue, S the distribution of service times for jobs, and c the number of servers at the node. For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (where inter-arrival durations are exponentially distributed) and have exponentially distributed service times (the M denotes a Markov process). In an M/G/1 queue, the G stands for "general" and indicates an arbitrary probability distribution for service times.

Example analysis of an M/M/1 queue
Consider a queue with one server and the following characteristics:
 * $$\lambda$$: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
 * $$\mu$$: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
 * n: the parameter characterizing the number of customers in the system
 * $$P_n$$: the probability of there being n customers in the system in steady state

Further, let $$E_n$$ represent the number of times the system enters state n, and $$L_n$$ represent the number of times the system leaves state n. Then $$\left\vert E_n - L_n \right\vert \in \{0, 1\}$$ for all n. That is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in the future ($$E_n = L_n$$) or not ($$\left\vert E_n - L_n \right\vert = 1$$).

When the system arrives at a steady state, the arrival rate should be equal to the departure rate.

Thus the balance equations
 * $$\mu P_1 = \lambda P_0$$
 * $$\lambda P_0 + \mu P_2 = (\lambda + \mu) P_1$$
 * $$\lambda P_{n-1} + \mu P_{n+1} = (\lambda + \mu) P_n$$

imply
 * $$P_n = \frac{\lambda}{\mu} P_{n-1},\ n=1,2,\ldots$$

The fact that $$P_0 + P_1 + \cdots = 1$$ leads to the geometric distribution formula
 * $$P_n = (1 - \rho) \rho^n$$

where $$\rho = \frac{\lambda}{\mu} < 1$$.

Simple two-equation queue
A common basic queueing system is attributed to Erlang and is a modification of Little's Law. Given an arrival rate λ, a dropout rate σ, and a departure rate μ, length of the queue L is defined as:

 * $$L = \frac{\lambda - \sigma}{\mu}$$.

Assuming an exponential distribution for the rates, the waiting time W can be defined as the proportion of arrivals that are served. This is equal to the exponential survival rate of those who do not drop out over the waiting period, giving:

 * $$\frac{\mu}{\lambda} = e^{-W{\mu}}$$

The second equation is commonly rewritten as:

 * $$W = \frac{1}{\mu} \mathrm{ln}\frac{\lambda}{\mu}$$

The two-stage one-box model is common in epidemiology.

History
In 1909, Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920. In Kendall's notation:

 * M stands for "Markov" or "memoryless", and means arrivals occur according to a Poisson process
 * D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service
 * k describes the number of servers at the queueing node (k = 1, 2, 3, ...)

If the node has more jobs than servers, then jobs will queue and wait for service.

The M/G/1 queue was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula.

After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953, David George Kendall solved the GI/M/k queue and introduced the modern notation for queues, now known as Kendall's notation. In 1957, Pollaczek studied the GI/G/1 using an integral equation. John Kingman gave a formula for the mean waiting time in a G/G/1 queue, now known as Kingman's formula.

Leonard Kleinrock worked on the application of queueing theory to message switching in the early 1960s and packet switching in the early 1970s. His initial contribution to this field was his doctoral thesis at the Massachusetts Institute of Technology in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the ARPANET, a forerunner to the Internet.

The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.

Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.

Modern day application of queueing theory concerns among other things product development where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.

Problems such as performance metrics for the M/G/k queue remain an open problem.

Service disciplines
Various scheduling policies can be used at queueing nodes:

 * First in, first out: Fifo queue.png Also called first-come, first-served (FCFS), this principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.

 * Last in, first out: This principle also serves customers one at a time, but the customer with the shortest waiting time will be served first. Also known as a stack.

 * Processor sharing: Service capacity is shared equally between customers.

 * Priority: Customers with high priority are served first. Priority queues can be of two types: non-preemptive (where a job in service cannot be interrupted) and preemptive (where a job in service can be interrupted by a higher-priority job). No work is lost in either model.

 * Shortest job first: The next job to be served is the one with the smallest size.

 * Preemptive shortest job first: The next job to be served is the one with the smallest original size.

 * Shortest remaining processing time: The next job to serve is the one with the smallest remaining processing requirement.

 * Service facility
 * Single server: customers line up and there is only one server
 * Several parallel servers (single queue): customers line up and there are several servers
 * Several parallel servers (several queues): there are many counters and customers can decide for which to queue

 * Unreliable server

Server failures occur according to a stochastic (random) process (usually Poisson) and are followed by setup periods during which the server is unavailable. The interrupted customer remains in the service area until server is fixed.

 * Customer waiting behavior
 * Balking: customers decide not to join the queue if it is too long
 * Jockeying: customers switch between queues if they think they will get served faster by doing so
 * Reneging: customers leave the queue if they have waited too long for service

Arriving customers not served (either due to the queue having no buffer, or due to balking or reneging by the customer) are also known as dropouts. The average rate of dropouts is a significant parameter describing a queue.

Queueing networks
Queue networks are systems in which multiple queues are connected by customer routing. When a customer is serviced at one node, it can join another node and queue for service, or leave the network.

For networks of m nodes, the state of the system can be described by an m–dimensional vector (x1, x2, ..., xm) where xi represents the number of customers at each node.

The simplest non-trivial networks of queues are called tandem queues. The first significant results in this area were Jackson networks, for which an efficient product-form stationary distribution exists and the mean value analysis (which allows average metrics such as throughput and sojourn times) can be computed. If the total number of customers in the network remains constant, the network is called a closed network and has been shown to also have a product–form stationary distribution by the Gordon–Newell theorem. This result was extended to the BCMP network, where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm, proposed in 1973.

Networks of customers have also been investigated, such as Kelly networks, where customers of different classes experience different priority levels at different service nodes. Another type of network are G-networks, first proposed by Erol Gelenbe in 1993: these networks do not assume exponential time distributions like the classic Jackson network.

Routing algorithms
In discrete-time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single-person service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput. A network scheduler must choose a queueing algorithm, which affects the characteristics of the larger network.

Mean-field limits
Mean-field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues m approaches infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.

Heavy traffic/diffusion approximations
In a system with high occupancy rates (utilisation near 1), a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion, Ornstein–Uhlenbeck process, or more general diffusion process. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative orthant.

Fluid limits
Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable but have an unstable fluid limit.

Queueing Applications
Queueing theory finds widespread application in computer science and information technology. In networking, for instance, queues are integral to routers and switches, where packets queue up for transmission. By applying queueing theory principles, designers can optimize these systems, ensuring responsive performance and efficient resource utilization. Beyond the technological realm, queueing theory is relevant to everyday experiences. Whether waiting in line at a supermarket or for public transportation, understanding the principles of queueing theory provides valuable insights into optimizing these systems for enhanced user satisfaction. At some point, everyone will be involved in an aspect of queuing. What some may view to be an inconvenience could possibly be the most effective method. Queueing theory, a discipline rooted in applied mathematics and computer science, is a field dedicated to the study and analysis of queues, or waiting lines, and their implications across a diverse range of applications. This theoretical framework has proven instrumental in understanding and optimizing the efficiency of systems characterized by the presence of queues. The study of queues is essential in contexts such as traffic systems, computer networks, telecommunications, and service operations. Queueing theory delves into various foundational concepts, with the arrival process and service process being central. The arrival process describes the manner in which entities join the queue over time, often modeled using stochastic processes like Poisson processes. The efficiency of queueing systems is gauged through key performance metrics. These include the average queue length, average wait time, and system throughput. These metrics provide insights into the system's functionality, guiding decisions aimed at enhancing performance and reducing wait times. References: Gross, D., & Harris, C. M. (1998). Fundamentals of Queueing Theory. John Wiley & Sons. Kleinrock, L. (1976). Queueing Systems: Volume I - Theory. Wiley. Cooper, B. F., & Mitrani, I. (1985). Queueing Networks: A Fundamental Approach. John Wiley & Sons

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Queueing theory


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Gaussian quadrature

[[File:Comparison Gaussquad trapezoidal.svg|thumb|upright=1.5|alt=Comparison between 2-point Gaussian and trapezoidal quadrature.|Comparison between 2-point Gaussian and trapezoidal quadrature.

The blue curve shows the function whose definite integral on the interval $[−1, 1]$ is to be calculated (the integrand). The trapezoidal rule approximates the function with a linear function that coincides with the integrand at the endpoints of the interval and is represented by an orange dashed line. The approximation is apparently not good, so the error is large (the trapezoidal rule gives an approximation of the integral equal to $y(–1) + y(1) = –10$, while the correct value is $2/3$). To obtain a more accurate result, the interval must be partitioned into many subintervals and then the composite trapezoidal rule must be used, which requires much more calculations.

The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the polynomial of degree 3 ($y(x) = 7x3 – 8x2 – 3x + 3$), the 2-point Gaussian quadrature rule even returns an exact result.]]

In numerical analysis, an $n$-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree $2n − 1$ or less by a suitable choice of the nodes $xi$ and weights $wi$ for $i = 1, ..., n$.

The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as $[−1, 1]$, so the rule is stated as $$\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i),$$

which is exact for polynomials of degree $2n − 1$ or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if $f (x)$ is well-approximated by a polynomial of degree $2n − 1$ or less on $[−1, 1]$.

The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as

$$f(x) = \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x),\quad \alpha,\beta > -1,$$

where $g(x)$ is well-approximated by a low-degree polynomial, then alternative nodes $xi'$ and weights $wi'$ will usually give more accurate quadrature rules. These are known as Gauss–Jacobi quadrature rules, i.e.,

$$\int_{-1}^1 f(x)\,dx = \int_{-1}^1 \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x)\,dx \approx \sum_{i=1}^n w_i' g\left(x_i'\right).$$

Common weights include $\frac{1}{\sqrt{1 - x^2}}$ (Chebyshev–Gauss) and $\sqrt{1 - x^2}$. One may also want to integrate over semi-infinite (Gauss–Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).

It can be shown (see Press et al., or Stoer and Bulirsch) that the quadrature nodes $xi$ are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.

Gauss–Legendre quadrature

For the simplest integration problem stated above, i.e., $n = 5)$ is well-approximated by polynomials on $$[-1, 1]$$, the associated orthogonal polynomials are Legendre polynomials, denoted by $f(x)$. With the $n$-th polynomial normalized to give $P_{n}(x)$, the $i$-th Gauss node, $x_{i}$, is the $i$-th root of $P_{n}$ and the weights are given by the formula $$ w_i = \frac{2}{\left( 1 - x_i^2 \right) \left[P'_n(x_i)\right]^2}.$$

Some low-order quadrature rules are tabulated below (over interval $P_{n}(1) = 1$, see the section below for other intervals).

Change of interval
An integral over $[−1, 1]$ must be changed into an integral over $[a, b]$ before applying the Gaussian quadrature rule. This change of interval can be done in the following way: $$\int_a^b f(x)\,dx = \int_{ -1}^1 f\left(\frac{b-a}{2}\xi + \frac{a+b}{2}\right)\,\frac{dx}{d\xi}d\xi$$

with $$\frac{dx}{d\xi} = \frac{b-a}{2}$$

Applying the $$n$$ point Gaussian quadrature $$(\xi, w)$$ rule then results in the following approximation: $$\int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}\xi_i + \frac{a+b}{2}\right).$$

Example of two-point Gauss quadrature rule
Use the two-point Gauss quadrature rule to approximate the distance in meters covered by a rocket from $$t = 8\mathrm{s} $$ to $$t = 30\mathrm{s},$$ as given by $$s = \int_{8}^{30}{\left( 2000\ln\left[ \frac{140000}{140000 - 2100t} \right] - 9.8t \right){dt}}$$

Change the limits so that one can use the weights and abscissae given in Table 1. Also, find the absolute relative true error. The true value is given as 11061.34 m.

Solution

First, changing the limits of integration from $$\left[ 8,30 \right]$$ to $$\left[ - 1,1 \right]$$ gives

$$ \begin{align} \int_{8}^{30} {f(t) dt} &= \frac{30 - 8}{2} \int_{- 1}^{1}{f\left( \frac{30 - 8}{2}x + \frac{30 + 8}{2} \right){dx}} \\ &= 11\int_{- 1}^{1}{f\left( 11x + 19 \right){dx}} \end{align} $$

Next, get the weighting factors and function argument values from Table 1 for the two-point rule,

 * $$c_1 = 1.000000000 $$
 * $$x_1 = - 0.577350269 $$
 * $$c_2 = 1.000000000 $$
 * $$x_2 = 0.577350269 $$

Now we can use the Gauss quadrature formula $$ \begin{align} 11\int_{-1}^{1}{f\left( 11x + 19 \right){dx}} & \approx 11\left[ c_1 f\left( 11 x_1 + 19 \right) + c_2 f\left( 11 x_2 + 19 \right) \right] \\ &= 11\left[ f\left( 11( - 0.5773503) + 19 \right) + f\left( 11(0.5773503) + 19 \right) \right] \\ &= 11\left[ f(12.64915) + f(25.35085) \right] \\ &= 11\left[ (296.8317) + (708.4811) \right] \\ &= 11058.44 \end{align}$$ since $$ \begin{align} f(12.64915) & = 2000\ln\left[ \frac{140000}{140000 - 2100(12.64915)} \right] - 9.8(12.64915) \\ &= 296.8317 \end{align}$$ $$ \begin{align} f(25.35085) & = 2000\ln\left[ \frac{140000}{140000 - 2100(25.35085)} \right] - 9.8(25.35085) \\ &= 708.4811 \end{align}$$

Given that the true value is 11061.34 m, the absolute relative true error, $$\left| \varepsilon_{t} \right|$$ is $$ \left| \varepsilon_{t} \right| = \left| \frac{11061.34 - 11058.44}{11061.34} \right| \times 100\% = 0.0262\% $$

Other forms
The integration problem can be expressed in a slightly more general way by introducing a positive weight function $n$ into the integrand, and allowing an interval other than $[−1, 1]$. That is, the problem is to calculate $$ \int_a^b \omega(x)\,f(x)\,dx $$ for some choices of $x_{i}$, $w_{i}$, and $ω$. For $[−1, 1]$, $a = −1$, and $b = 1$, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A & S).

Fundamental theorem
Let $a$ be a nontrivial polynomial of degree $b$ such that $$\int_a^b \omega(x) \, x^k p_n(x) \, dx = 0, \quad \text{for all } k = 0, 1, \ldots, n - 1.$$

Note that this will be true for all the orthogonal polynomials above, because each $ω$ is constructed to be orthogonal to the other polynomials $[−1, 1]$ for $ω(x) = 1$, and $ω(x)$ is in the span of that set.

If we pick the $(−1, 1)$ nodes $(−1, 1)$ to be the zeros of $[−1, 1]$, then there exist $[0, ∞)$ weights $[0, ∞)$ which make the Gaussian quadrature computed integral exact for all polynomials $1$ of degree $β = 0$ or less. Furthermore, all these nodes $(−∞, ∞)$ will lie in the open interval $j<n$.

To prove the first part of this claim, let $x^{k}$ be any polynomial of degree $h(x)$ or less. Divide it by the orthogonal polynomial $p_{n}$ to get $$ h(x) = p_n(x) \, q(x) + r(x). $$ where $2n − 1$ is the quotient, of degree $(a, b)$ or less (because the sum of its degree and that of the divisor $n$ must equal that of the dividend), and $h(x)$ is the remainder, also of degree $2n − 1$ or less (because the degree of the remainder is always less than that of the divisor). Since $p_{n}$ is by assumption orthogonal to all monomials of degree less than $p_{j}$, it must be orthogonal to the quotient $q(x)$. Therefore $$ \int_a^b \omega(x)\,h(x)\,dx = \int_a^b \omega(x)\,\big( \, p_n(x) q(x) + r(x) \, \big)\,dx = \int_a^b \omega(x)\,r(x)\,dx. $$

Since the remainder $n − 1$ is of degree $r(x)$ or less, we can interpolate it exactly using $n$ interpolation points with Lagrange polynomials $n − 1$, where $$ l_i(x) = \prod _{j \ne i} \frac{x-x_j}{x_i-x_j}. $$

We have $$ r(x) = \sum_{i=1}^n l_i(x) \, r(x_i). $$

Then its integral will equal $$ \int_a^b \omega(x)\,r(x)\,dx = \int_a^b \omega(x) \, \sum_{i=1}^n l_i(x) \, r(x_i) \, dx = \sum_{i=1}^n \, r(x_i) \, \int_a^b \omega(x) \, l_i(x) \, dx = \sum_{i=1}^n \, r(x_i) \, w_i, $$

where $q(x)$, the weight associated with the node $r(x)$, is defined to equal the weighted integral of $n − 1$ (see below for other formulas for the weights). But all the $x_{i}$ are roots of $p_{n}$, so the division formula above tells us that $$ h(x_i) = p_n(x_i) \, q(x_i) + r(x_i) = r(x_i), $$ for all $n$. Thus we finally have $$ \int_a^b \omega(x)\,h(x)\,dx = \int_a^b \omega(x) \, r(x) \, dx = \sum_{i=1}^n w_i \, r(x_i) = \sum_{i=1}^n w_i \, h(x_i). $$

This proves that for any polynomial $l_{i}(x)$ of degree $w_{i}$ or less, its integral is given exactly by the Gaussian quadrature sum.

To prove the second part of the claim, consider the factored form of the polynomial $x_{i}$. Any complex conjugate roots will yield a quadratic factor that is either strictly positive or strictly negative over the entire real line. Any factors for roots outside the interval from $w_{i}$ to $x_{i}$ will not change sign over that interval. Finally, for factors corresponding to roots $p_{n}$ inside the interval from $p_{n}$ to $p_{n}$ that are of odd multiplicity, multiply $l_{i}(x)$ by one more factor to make a new polynomial $$ p_n(x) \, \prod_i (x - x_i). $$

This polynomial cannot change sign over the interval from $n$ to $n$ because all its roots there are now of even multiplicity. So the integral $$ \int_a^b p_n(x) \, \left( \prod_i (x - x_i) \right) \, \omega(x) \, dx \ne 0, $$ since the weight function $h(x)$ is always non-negative. But $2n − 1$ is orthogonal to all polynomials of degree $p_{n}$ or less, so the degree of the product $$ \prod_i (x - x_i) $$ must be at least $x_{i}$. Therefore $p_{n}$ has $p_{n}$ distinct roots, all real, in the interval from $i$ to $a$.

General formula for the weights
The weights can be expressed as

where $$a_{k}$$ is the coefficient of $$x^{k}$$ in $$p_{k}(x)$$. To prove this, note that using Lagrange interpolation one can express $ω(x)$ in terms of $$r(x_{i})$$ as $$r(x) = \sum_{i=1}^{n} r(x_{i}) \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}$$ because $p_{n}$ has degree less than $b$ and is thus fixed by the values it attains at $x_{i}$ different points. Multiplying both sides by $n-1$ and integrating from $a$ to $b$ yields $$\int_{a}^{b}\omega(x)r(x)dx = \sum_{i=1}^{n} r(x_{i}) \int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}} \frac{x-x_{j}}{x_{i}-x_{j}}dx$$

The weights $a$ are thus given by $$w_{i} = \int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx$$

This integral expression for $$w_{i}$$ can be expressed in terms of the orthogonal polynomials $$p_{n}(x)$$ and $$p_{n-1}(x)$$ as follows.

We can write $$ \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}} \left(x-x_{j}\right) = \frac{\prod_{1\leq j\leq n} \left(x - x_{j}\right)}{x-x_{i}} = \frac{p_{n}(x)}{a_{n}\left(x-x_{i}\right)}$$

where $$a_{n}$$ is the coefficient of $$x^n$$ in $$p_{n}(x)$$. Taking the limit of $b$ to $$x_{i}$$ yields using L'Hôpital's rule $$ \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}} \left(x_{i}-x_{j}\right) = \frac{p'_{n}(x_{i})}{a_{n}}$$

We can thus write the integral expression for the weights as

In the integrand, writing $$\frac{1}{x-x_i} = \frac{1 - \left(\frac{x}{x_i}\right)^{k}}{x - x_i} + \left(\frac{x}{x_i}\right)^{k} \frac{1}{x - x_i}$$

yields $$\int_a^b\omega(x)\frac{x^kp_n(x)}{x-x_i}dx = x_i^k \int_{a}^{b}\omega(x)\frac{p_n(x)}{x-x_i}dx$$

provided $$k \leq n$$, because $$\frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}}$$ is a polynomial of degree $p_{n}$ which is then orthogonal to $$p_{n}(x)$$. So, if $r(x)$ is a polynomial of at most nth degree we have $$\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}} dx = \frac{1}{q(x_{i})} \int_{a}^{b} \omega(x)\frac{q(x) p_n(x)}{x-x_{i}}dx $$

We can evaluate the integral on the right hand side for $$q(x) = p_{n-1}(x)$$ as follows. Because $$\frac{p_{n}(x)}{x-x_{i}}$$ is a polynomial of degree $r(x)$, we have $$\frac{p_{n}(x)}{x-x_{i}} = a_{n}x^{n-1} + s(x)$$ where $ω(x)$ is a polynomial of degree $$n - 2$$. Since $k − 1$ is orthogonal to $$p_{n-1}(x)$$ we have $$\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{p_{n-1}(x_{i})} \int_{a}^{b}\omega(x)p_{n-1}(x)x^{n-1}dx $$

We can then write $$x^{n-1} = \left(x^{n-1} - \frac{p_{n-1}(x)}{a_{n-1}}\right) + \frac{p_{n-1}(x)}{a_{n-1}}$$

The term in the brackets is a polynomial of degree $$n - 2$$, which is therefore orthogonal to $$p_{n-1}(x)$$. The integral can thus be written as $$\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx = \frac{a_{n}}{a_{n-1} p_{n-1}(x_{i})} \int_{a}^{b}\omega(x) p_{n-1}(x)^{2} dx $$

According to equation ($n$), the weights are obtained by dividing this by $$p'_{n}(x_{i})$$ and that yields the expression in equation ($n$).

$$w_{i}$$ can also be expressed in terms of the orthogonal polynomials $$p_{n}(x)$$ and now $$p_{n+1}(x)$$. In the 3-term recurrence relation $$p_{n+1}(x_{i}) = (a) p_{n}(x_{i}) + (b) p_{n-1}(x_{i})$$ the term with $$p_{n}(x_{i})$$ vanishes, so $$p_{n-1}(x_{i})$$ in Eq. (1) can be replaced by $\frac{1}{b} p_{n+1} \left(x_i\right)$.

Proof that the weights are positive
Consider the following polynomial of degree $$2n - 2$$ $$f(x) = \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}}\frac{\left(x - x_j\right)^2}{\left(x_i - x_j\right)^2}$$ where, as above, the $a$ are the roots of the polynomial $$p_{n}(x)$$. Clearly $$f(x_j) = \delta_{ij}$$. Since the degree of $$f(x)$$ is less than $$2n - 1$$, the Gaussian quadrature formula involving the weights and nodes obtained from $$p_{n}(x)$$ applies. Since $$f(x_{j}) = 0$$ for $b$ not equal to $$, we have $$\int_{a}^{b}\omega(x)f(x)dx=\sum_{j=1}^{n}w_{j}f(x_{j}) = \sum_{j=1}^{n} \delta_{ij} w_j = w_{i} > 0.$$

Since both $$\omega(x)$$ and $$f(x)$$ are non-negative functions, it follows that $$w_{i} > 0$$.

Computation of Gaussian quadrature rules
There are many algorithms for computing the nodes $n$ and weights $n$ of Gaussian quadrature rules. The most popular are the Golub-Welsch algorithm requiring $q(x)$ operations, Newton's method for solving $$p_n(x) = 0$$ using the three-term recurrence for evaluation requiring $n − 1$ operations, and asymptotic formulas for large n requiring $s(x)$ operations.

Recurrence relation
Orthogonal polynomials $$p_r$$ with $$(p_r, p_s) = 0$$ for $$r \ne s$$ for a scalar product $$(\cdot, \cdot)$$, degree $$(p_r) = r$$ and leading coefficient one (i.e. monic orthogonal polynomials) satisfy the recurrence relation $$p_{r+1}(x) = (x - a_{r,r}) p_r(x) - a_{r,r-1} p_{r-1}(x) \cdots - a_{r,0}p_0(x)$$

and scalar product defined $$(f(x),g(x))=\int_a^b\omega(x)f(x)g(x)dx$$

for $$r = 0, 1, \ldots, n - 1$$ where $a$ is the maximal degree which can be taken to be infinity, and where $a_{r,s} = \frac{\left(xp_r, p_s\right)}{\left(p_s, p_s\right)}$. First of all, the polynomials defined by the recurrence relation starting with $$p_0(x) = 1$$ have leading coefficient one and correct degree. Given the starting point by $$p_0$$, the orthogonality of $$p_r$$ can be shown by induction. For $$r = s = 0$$ one has $$(p_1,p_0) = (x-a_{0,0}) (p_0,p_0) = (xp_0,p_0) - a_{0,0}(p_0,p_0) = (xp_0,p_0) - (xp_0,p_0) = 0.$$

Now if $$p_0, p_1, \ldots, p_r$$ are orthogonal, then also $$p_{r+1}$$, because in $$(p_{r+1}, p_s) = (xp_r, p_s) - a_{r,r}(p_r, p_s) - a_{r,r-1}(p_{r-1}, p_s)\cdots - a_{r,0}(p_0, p_s)$$ all scalar products vanish except for the first one and the one where $$p_s$$ meets the same orthogonal polynomial. Therefore, $$(p_{r+1},p_s) = (xp_r,p_s) - a_{r,s}(p_s,p_s) = (xp_r,p_s)-(xp_r,p_s) = 0.$$

However, if the scalar product satisfies $$(xf, g) = (f,xg)$$ (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For $$s < r - 1, xp_s$$ is a polynomial of degree less than or equal to $s(x)$. On the other hand, $$p_r$$ is orthogonal to every polynomial of degree less than or equal to $O(n^{2})$. Therefore, one has $$(xp_r, p_s) = (p_r, xp_s) = 0$$ and $$a_{r,s} = 0$$ for $O(n^{2})$. The recurrence relation then simplifies to $$p_{r+1}(x) = (x-a_{r,r}) p_r(x) - a_{r,r-1} p_{r-1}(x)$$

or $$p_{r+1}(x) = (x-a_r) p_r(x) - b_r p_{r-1}(x)$$

(with the convention $$p_{-1}(x) \equiv 0$$) where $$a_r := \frac{(xp_r,p_r)}{(p_r,p_r)}, \qquad b_r := \frac{(xp_r,p_{r-1})}{(p_{r-1},p_{r-1})} = \frac{(p_r,p_r)}{(p_{r-1},p_{r-1})}$$

(the last because of $$(xp_r, p_{r-1}) = (p_r, xp_{r-1}) = (p_r, p_r)$$, since $$xp_{r-1}$$ differs from $$p_r$$ by a degree less than $b$).

The Golub-Welsch algorithm
The three-term recurrence relation can be written in matrix form $$J\tilde{P} = x\tilde{P} - p_n(x) \mathbf{e}_n$$ where $$\tilde{P} = \begin{bmatrix} p_0(x) & p_1(x) & \cdots & p_{n-1}(x) \end{bmatrix}^\mathsf{T}$$, $$\mathbf{e}_n$$ is the $$n$$th standard basis vector, i.e., $$\mathbf{e}_n = \begin{bmatrix} 0 & \cdots & 0 & 1 \end{bmatrix}^\mathsf{T}$$, and $w_{i}$ is the following tridiagonal matrix, called the Jacobi matrix: $$\mathbf{J} = \begin{bmatrix} a_0 &     1 &      0 &  \cdots  &       0 \\ b_1 &   a_1 &      1 &  \ddots  &  \vdots \\ 0 &   b_2 & \ddots &  \ddots  &       0 \\ \vdots & \ddots & \ddots & a_{n-2} &       1 \\ 0 & \cdots &      0 & b_{n-1} & a_{n-1} \end{bmatrix}.$$

The zeros $$x_j$$ of the polynomials up to degree $x$, which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this matrix. This procedure is known as Golub–Welsch algorithm.

For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix $$\mathcal{J}$$ with elements $$\begin{align} \mathcal{J}_{k,i} = J_{k,i} &= a_{k-1}       & k &= 1,2,\ldots,n \\[2.1ex] \mathcal{J}_{k-1,i} = \mathcal{J}_{k,k-1} = \sqrt{J_{k,k-1}J_{k-1,k}} &= \sqrt{b_{k-1}} & k &= \hphantom{1,\,}2,\ldots,n. \end{align}$$

That is,

$$\mathcal{J} = \begin{bmatrix} a_0 & \sqrt{b_1} &         0 &        \cdots  &              0 \\ \sqrt{b_1} &       a_1 & \sqrt{b_2} &        \ddots  &         \vdots \\ 0 & \sqrt{b_2} &    \ddots &        \ddots  &              0 \\ \vdots &    \ddots &     \ddots &        a_{n-2} & \sqrt{b_{n-1}} \\ 0 &    \cdots &          0 & \sqrt{b_{n-1}} &        a_{n-1} \end{bmatrix}.$$

$O(n)$ and $$\mathcal{J}$$ are similar matrices and therefore have the same eigenvalues (the nodes). The weights can be computed from the corresponding eigenvectors: If $$\phi^{(j)}$$ is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated with the eigenvalue $$, the corresponding weight can be computed from the first component of this eigenvector, namely: $$w_j = \mu_0 \left(\phi_1^{(j)}\right)^2$$

where $$\mu_0$$ is the integral of the weight function $$\mu_0 = \int_a^b \omega(x) dx.$$

See, for instance, for further details.

Error estimates
The error of a Gaussian quadrature rule can be stated as follows. For an integrand which has $r − 1$ continuous derivatives, $$ \int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i) = \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n, p_n) $$ for some $$ in $r − 1$, where $$ is the monic (i.e. the leading coefficient is $s < r − 1$) orthogonal polynomial of degree $x_{j}$ and where $$ (f,g) = \int_a^b \omega(x) f(x) g(x) \, dx.$$

In the important special case of $J$, we have the error estimate $$ \frac{\left(b - a\right)^{2n+1} \left(n!\right)^4}{(2n + 1)\left[\left(2n\right)!\right]^3} f^{(2n)} (\xi), \qquad a < \xi < b.$$

Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the order $2n$ derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss–Kronrod quadrature rules can be useful.

Gauss–Kronrod rules
If the interval $(a, b)$ is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers), and thus the integrand must be evaluated at every point. Gauss–Kronrod rules are extensions of Gauss quadrature rules generated by adding $1$ points to an $j$-point rule in such a way that the resulting rule is of order $ω(x) = 1$. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension is often used as an estimate of the approximation error.

Gauss–Lobatto rules
Also known as Lobatto quadrature, named after Dutch mathematician Rehuel Lobatto. It is similar to Gaussian quadrature with the following differences:
 * 1) The integration points include the end points of the integration interval.
 * 2) It is accurate for polynomials up to degree $2n$, where $i$ is the number of integration points.

Lobatto quadrature of function $[a, b]$ on interval $n + 1$: $$\int_{-1}^1 {f(x) \, dx} = \frac {2} {n(n-1)}[f(1) + f(-1)] + \sum_{i = 2}^{n-1} {w_i f(x_i)} + R_n.$$

Abscissas: $x_{i}$ is the $$(i - 1)$$st zero of $$P'_{n-1}(x)$$, here $$P_m(x)$$ denotes the standard Legendre polynomial of $w_{i}$-th degree and the dash denotes the derivative.

Weights: $$w_i = \frac{2}{n(n - 1)\left[P_{n-1}\left(x_i\right)\right]^2}, \qquad x_i \ne \pm 1.$$

Remainder: $$R_n = \frac{-n\left(n - 1\right)^3 2^{2n-1} \left[\left(n - 2\right)!\right]^4}{(2n-1) \left[\left(2n - 2\right)!\right]^3} f^{(2n-2)}(\xi), \qquad -1 < \xi < 1.$$

Some of the weights are:

An adaptive variant of this algorithm with 2 interior nodes is found in GNU Octave and MATLAB as  and.

title: Gaussian quadrature


text:
Orange County, Texas

Orange County is a county located in the very southeastern corner of the U.S. state of Texas, sharing a boundary with Louisiana, within the Golden Triangle of Texas. As of the 2020 census, its population was 84,808. The county seat is the city of Orange, and it falls within the Beaumont–Port Arthur metropolitan area.

History
Orange County was formed in 1852 from portions of Jefferson County. It was named after the orange fruit, the common citrus fruit grown by the early settlers of this county near the mouth of the Sabine River. Due to periodic spells of quite cold winter weather (frosts) in Orange County, it is no longer the home of orange trees and citrus orchards. The production of those fruits in Texas long ago was moved a long way southwest into the Rio Grande Valley, where the weather is almost always warm all winter long. Citrus trees produce their fruit in the wintertime, which makes them especially vulnerable to frost and icy weather.

A similar thing has happened in Florida, where orchards of citrus trees no longer exist in either Citrus County or Orange County because of bad winter freezes in some years. In both Florida and Texas, the citrus agriculture has been moved farther south in search of milder winters, and away from the periodic frosts.

During World War II, Orange County was the home of a large amount of shipbuilding for the navies the United States and allied countries. The major shipbuilder, Consolidated Steel Corporation, was located in the town of Orange, and among the warships that it built were the USS Aulick (DD-569) (1942), the first warship built there, the USS Pope (DE-134) (1943), and the USS Carpenter (DD-825) (1945–46), the last warship built there. During the war, the Consolidate Steel Corporation employed as many as 20,000 people at its shipyard in Orange.

Geography
According to the U.S. Census Bureau, the county has a total area of 380 sqmi, of which 334 sqmi are land and 46 sqmi (12%) are covered by water.

Orange County is bordered on its east by the Sabine River, on its southeast by Sabine Lake, and on the northwest by the Neches River.

The geography of Orange County varies relatively little, with an elevation that reaches 33 ft above sea level at very few points within the county. Orange County is very flat, and its soil is quite sandy, as could be expected in a county along the Gulf of Mexico. (Sandy soil is also common in southern Louisiana, Mississippi, and Alabama, and in western and southern Florida.) Saltwater marshes occur in much of the southeastern part of Orange County that borders the Sabine River. The Piney Woods are in the northern part of the county.

Adjacent counties and parishes

 * Jasper County (north)
 * Newton County (north)
 * Hardin County (northwest)
 * Jefferson County (west)
 * Calcasieu Parish, Louisiana (east)
 * Cameron Parish, Louisiana (southeast)

National protected area

 * Big Thicket National Preserve (part)

Cities

 * Bridge City
 * Orange (county seat)
 * Pine Forest
 * Pinehurst
 * Port Arthur (mostly in Jefferson County)
 * Rose City
 * Vidor
 * West Orange

Census-designated place

 * Mauriceville

Unincorporated communities

 * Forest Heights
 * Little Cypress
 * Orangefield

Ghost towns

 * Lemonville
 * Texla

Demographics
''Note: the US Census treats Hispanic/Latino as an ethnic category. This table excludes Latinos from the racial categories and assigns them to a separate category. Hispanics/Latinos can be of any race.''

As of the census of 2000, 84,966 people, 31,642 households, and 23,794 families resided in the county. The population density was 238 /mi2. The 34,781 housing units averaged 98 /mi2. The racial makeup of the county was 87.98% White, 8.38% African American, 0.56% Native American, 0.78% Asian, 1.15% from other races, and 1.15% from two or more races. About 3.62% of the population was Hispanic or Latino of any race.

Of the 31,642 households, 35.30% had children under the age of 18 living with them, 58.80% were married couples living together, 12.10% had a female householder with no husband present, and 24.80% were not families. About 21.7% of all households were made up of individuals, and 9.30% had someone living alone who was 65 years of age or older. The average household size was 2.65 and the average family size was 3.08.

In the county, the population was distributed as 27.30% under the age of 18, 8.70% from 18 to 24, 28.10% from 25 to 44, 23.20% from 45 to 64, and 12.70% who were 65 years of age or older. The median age was 36 years. For every 100 females, there were 96.40 males. For every 100 females age 18 and over, there were 92.60 males.

The median income for a household in the county was $37,586, and for a family was $44,152. Males had a median income of $40,185 versus $21,859 for females. The per capita income for the county was $17,554. About 11.40% of families and 13.80% of the population were below the poverty line, including 18.50% of those under age 18 and 12.40% of those age 65 or over.

Government
The Orange County Courthouse serves as the court for the region. Republican County Judge John Gothia presides over the five-member Orange County Commissioners' Court.

Orange County lies in Texas House District 21, represented beginning in 2015 by Republican Dade Phelan of Beaumont.

Economy
Primary economic activities in Orange County are the petroleum refining industry, paper milling, rice farming, and shrimping.

Orange County was formerly a center for the building of warships, and a large U.S. Navy ghost fleet (reserve fleet) still exists in Jefferson County - from which currently, many old warships are being cleaned of water pollution sources and then scrapped for their metals, thus employment for residents of Orange County in shipbreaking.

Newspapers published in the county include the twice-weekly Orange Leader and weeklies including the Bridge City-based Penny Record, County Record, and Vidor Vidorian.

Airports
Orange County Airport operates general-aviation flights.

Nearby Southeast Texas Regional Airport (Port Arthur) operates commercial flights.

Major highways

 * I-10 (TX).svg Interstate 10
 * US 90.svg U.S. Highway 90
 * Texas 12.svg State Highway 12
 * Texas 62.svg State Highway 62
 * Texas 73.svg State Highway 73
 * Texas 87.svg State Highway 87

Education
The county is served by five school districts: Bridge City ISD,  Little Cypress-Mauriceville Consolidated ISD,  Orangefield ISD,  Vidor ISD, and West Orange-Cove Consolidated ISD. <!--

West Orange-Cove Consolidated Independent School District
The district's Superintendent of Schools is James Colbert Jr.
 * M B North Early Learning Center/Head Start (seeking, Principal)
 * West Orange-Stark Elementary (Bennie Smith, Principal)
 * West Orange-Stark Middle (seeking, Principal)
 * West Orange-Stark High School (Hutcherson Hill, Principal)
 * West Orange-Cove Academic Alternative Center (Rodney Anderson, Director)

Bridge City ISD
The Superintendent is Jamey Harrison. It includes:
 * Hatton Elementary School (Principal:Norman Gaspard )
 * Sims Elementary School (Principal:Kent Broussard )
 * Bridge City Intermediate School (Principal:Tara Fountain )
 * Bridge City Middle School (Principal:Kevin Jones )
 * Bridge City High School (Principal:Richard Briggs ).

Little Cypress-Mauriceville Consolidated ISD
The Superintendent is Dr. Pauline Hargrove. It includes: -->
 * Mauriceville Elementary School
 * Little Cypress Elementary School
 * Mauriceville Middle School
 * Little Cypress Intermediate School
 * Mauriceville Middle School
 * Little Cypress Junior High School
 * Little Cypress-Mauriceville High School

title: Orange County, Texas


text:
Zircon

Zircon   is a mineral belonging to the group of nesosilicates and is a source of the metal zirconium. Its chemical name is zirconium(IV) silicate, and its corresponding chemical formula is ZrSiO4. An empirical formula showing some of the range of substitution in zircon is (Zr1–y, REEy)(SiO4)1–x(OH)4x–y. Zircon precipitates from silicate melts and has relatively high concentrations of high field strength incompatible elements. For example, hafnium is almost always present in quantities ranging from 1 to 4%. The crystal structure of zircon is tetragonal crystal system. The natural color of zircon varies between colorless, yellow-golden, red, brown, blue, and green.

The name derives from the Persian zargun, meaning "gold-hued". This word is changed into "jargoon", a term applied to light-colored zircons. The English word "zircon" is derived from Zirkon, which is the German adaptation of this word. Yellow, orange, and red zircon is also known as "hyacinth", from the flower hyacinthus, whose name is of Ancient Greek origin.

Properties

Zircon is common in the crust of Earth. It occurs as a common accessory mineral in igneous rocks (as primary crystallization products), in metamorphic rocks and as detrital grains in sedimentary rocks. Large zircon crystals are rare. Their average size in granite rocks is about 0.1 –, but they can also grow to sizes of several cm, especially in mafic pegmatites and carbonatites. Zircon is fairly hard (with a Mohs hardness of 7.5) and chemically stable, and so is highly resistant to weathering. It also is resistant to heat, so that detrital zircon grains are sometimes preserved in igneous rocks formed from melted sediments. Its resistance to weathering, together with its relatively high specific gravity (4.68), make it an important component of the heavy mineral fraction of sandstones.

Because of their uranium and thorium content, some zircons undergo metamictization. Connected to internal radiation damage, these processes partially disrupt the crystal structure and partly explain the highly variable properties of zircon. As zircon becomes more and more modified by internal radiation damage, the density decreases, the crystal structure is compromised, and the color changes.

Zircon occurs in many colors, including reddish brown, yellow, green, blue, gray, and colorless. The color of zircons can sometimes be changed by heat treatment. Common brown zircons can be transformed into colorless and blue zircons by heating to 800 to 1000 C. In geological settings, the development of pink, red, and purple zircon occurs after hundreds of millions of years, if the crystal has sufficient trace elements to produce color centers. Color in this red or pink series is annealed in geological conditions above temperatures of around 400 C.

Structurally, zircon consists of parallel chains of alternating silica tetrahedra (silicon ions in fourfold coordination with oxygen ions) and zirconium ions, with the large zirconium ions in eightfold coordination with oxygen ions.

Applications
Zircon is mainly consumed as an opacifier, and has been known to be used in the decorative ceramics industry. It is also the principal precursor not only to metallic zirconium, although this application is small, but also to all compounds of zirconium including zirconium dioxide (ZrO2), an important refractory oxide with a melting point of 2717 C.

Other applications include use in refractories and foundry casting and a growing array of specialty applications as zirconia and zirconium chemicals, including in nuclear fuel rods, catalytic fuel converters and in water and air purification systems.

Zircon is one of the key minerals used by geologists for geochronology.

Zircon is a part of the ZTR index to classify highly-weathered sediments.

Gemstone
Transparent zircon is a well-known form of semi-precious gemstone, favored for its high specific gravity (between 4.2 and 4.86) and adamantine luster. Because of its high refractive index (1.92) it has sometimes been used as a substitute for diamond, though it does not display quite the same play of color as a diamond. Zircon is one of the heaviest types of gemstone. Its Mohs hardness is between that of quartz and topaz, at 7.5 on the 10 point scale, though below that of the similar manmade stone cubic zirconia (8-8.5). Zircons may sometimes lose their inherent color after long exposure to bright sunlight, which is unusual in a gemstone. It is immune to acid attack except by sulfuric acid and then only when ground into a fine powder.

Most gem-grade zircons show a high degree of birefringence which, on stones cut with a table and pavilion cuts (i.e., nearly all cut stones), can be seen as the apparent doubling-up of the latter when viewed through the former, and this characteristic can be used to distinguish them from diamonds and cubic zirconias (CZ) as well as soda-lime glass, none of which show this characteristic. However, some zircons from Sri Lanka display only weak or no birefringence at all, and some other Sri Lanka stones may show clear birefringence in one place and little or none in another part of the same cut stone. Other gemstones also display birefringence, so while the presence of this characteristic may help distinguish a given zircon from a diamond or a CZ, it will not help distinguish it from, for example, a topaz gemstone. The high specific gravity of zircon, however, can usually separate it from any other gem and is simple to test.

Also, birefringence depends on the cut of the stone in relation to its optical axis. If a zircon is cut with this axis perpendicular to its table, birefringence may be reduced to undetectable levels unless viewed with a jeweler's loupe or other magnifying optics. The highest grade zircons are cut to minimize birefringence.

The value of a zircon gem depends largely on its color, clarity, and size. Prior to World War II, blue zircons (the most valuable color) were available from many gemstone suppliers in sizes between 15 and 25 carats; since then, stones even as large as 10 carats have become very scarce, especially in the most desirable color varieties.

Synthetic zircons have been created in laboratories. They are occasionally used in jewellery such as earrings. Zircons are sometimes imitated by spinel and synthetic sapphire, but are not difficult to distinguish from them with simple tools.

Occurrence
Zircon is a common accessory to trace mineral constituent of all kinds of igneous rocks, but particularly granite and felsic igneous rocks. Due to its hardness, durability and chemical inertness, zircon persists in sedimentary deposits and is a common constituent of most sands. Zircon can occasionally be found as a trace mineral in ultrapotassic igneous rocks such as kimberlites, carbonatites, and lamprophyre, owing to the unusual magma genesis of these rocks.

Zircon forms economic concentrations within heavy mineral sands ore deposits, within certain pegmatites, and within some rare alkaline volcanic rocks, for example the Toongi Trachyte, Dubbo, New South Wales Australia in association with the zirconium-hafnium minerals eudialyte and armstrongite.

Australia leads the world in zircon mining, producing 37% of the world total and accounting for 40% of world EDR (economic demonstrated resources) for the mineral. South Africa is Africa's main producer, with 30% of world production, second after Australia.

Radiometric dating
Zircon has played an important role during the evolution of radiometric dating. Zircons contain trace amounts of uranium and thorium (from 10 ppm up to 1 wt%) and can be dated using several modern analytical techniques. Because zircons can survive geologic processes like erosion, transport, even high-grade metamorphism, they contain a rich and varied record of geological processes. Currently, zircons are typically dated by uranium-lead (U-Pb), fission-track, and U+Th/He techniques. Imaging the cathodoluminescence emission from fast electrons can be used as a prescreening tool for high-resolution secondary-ion-mass spectrometry (SIMS) to image the zonation pattern and identify regions of interest for isotope analysis. This is done using an integrated cathodoluminescence and scanning electron microscope. Zircons in sedimentary rock can identify the sediment source.

Zircons from Jack Hills in the Narryer Gneiss Terrane, Yilgarn Craton, Western Australia, have yielded U-Pb ages up to 4.404 billion years, interpreted to be the age of crystallization, making them the oldest minerals so far dated on Earth. In addition, the oxygen isotopic compositions of some of these zircons have been interpreted to indicate that more than 4.3 billion years ago there was already liquid water on the surface of the Earth. This interpretation is supported by additional trace element data, but is also the subject of debate. In 2015, "remains of biotic life" were found in 4.1-billion-year-old rocks in the Jack Hills of Western Australia. According to one of the researchers, "If life arose relatively quickly on Earth ... then it could be common in the universe."

Similar minerals
Hafnon (HfSiO4), xenotime (YPO4), béhierite, schiavinatoite ((Ta,Nb)BO4), thorite (ThSiO4), and coffinite (USiO4) all share the same crystal structure (IVX IVY O4, IIIX VY O4 in the case of xenotime) as zircon.

title: Zircon