# 4.9: Reduction of Order

In this section we give a method for finding the general solution of

[label{eq:5.6.1} P_0(x)y''+P_1(x)y'+P_2(x)y=F(x)]

if we know a nontrivial solution (y_1) of the complementary equation

[label{eq:5.6.2} P_0(x)y''+P_1(x)y'+P_2(x)y=0.]

The method is called reduction of order because it reduces the task of solving Equation ef{eq:5.6.1} to solving a first order equation. Unlike the method of undetermined coefficients, it does not require (P_0), (P_1), and (P_2) to be constants, or (F) to be of any special form.

By now you shoudn’t be surprised that we look for solutions of Equation ef{eq:5.6.1} in the form

[label{eq:5.6.3} y=uy_1]

where (u) is to be determined so that (y) satisfies Equation ef{eq:5.6.1}. Substituting Equation ef{eq:5.6.3} and

[egin{align*} y'&= u'y_1+uy_1' [4pt] y'' &= u''y_1+2u'y_1'+uy_1'' end{align*}]

into Equation ef{eq:5.6.1} yields

[P_0(x)(u''y_1+2u'y_1'+uy_1'')+P_1(x)(u'y_1+uy_1')+P_2(x)uy_1=F(x). onumber]

Collecting the coefficients of (u), (u'), and (u'') yields

[label{eq:5.6.4} (P_0y_1)u''+(2P_0y_1'+P_1y_1)u'+(P_0y_1''+P_1y_1'+P_2y_1) u=F.]

However, the coefficient of (u) is zero, since (y_1) satisfies Equation ef{eq:5.6.2}. Therefore Equation ef{eq:5.6.4} reduces to

[label{eq:5.6.5} Q_0(x)u''+Q_1(x)u'=F,]

with

(It isn’t worthwhile to memorize the formulas for (Q_0) and (Q_1)!) Since Equation ef{eq:5.6.5} is a linear first order equation in (u'), we can solve it for (u') by variation of parameters as in Section 1.2, integrate the solution to obtain (u), and then obtain (y) from Equation ef{eq:5.6.3}.

Example (PageIndex{1})

1. Find the general solution of [label{eq:5.6.6} xy''-(2x+1)y'+(x+1)y=x^2,] given that (y_1=e^x) is a solution of the complementary equation [label{eq:5.6.7} xy''-(2x+1)y'+(x+1)y=0.]
2. As a byproduct of (a), find a fundamental set of solutions of Equation ef{eq:5.6.7}.

Solution

a. If (y=ue^x), then (y'=u'e^x+ue^x) and (y''=u''e^x+2u'e^x+ue^x), so

[egin{align*} xy''-(2x+1)y'+(x+1)y&=x(u''e^x+2u'e^x+ue^x) -(2x+1)(u'e^x+ue^x)+(x+1)ue^x &=(xu''-u')e^x.end{align*}]

Therefore (y=ue^x) is a solution of Equation ef{eq:5.6.6} if and only if

[(xu''-u')e^x=x^2, onumber]

which is a first order equation in (u'). We rewrite it as

[label{eq:5.6.8} u''-{u'over x}=xe^{-x}.]

To focus on how we apply variation of parameters to this equation, we temporarily write (z=u'), so that Equation ef{eq:5.6.8} becomes

[label{eq:5.6.9} z'-{zover x}=xe^{-x}.]

We leave it to you to show (by separation of variables) that (z_1=x) is a solution of the complementary equation

[z'-{zover x}=0 onumber]

for Equation ef{eq:5.6.9}. By applying variation of parameters as in Section 1.2, we can now see that every solution of Equation ef{eq:5.6.9} is of the form

Since (u'=z=vx), (u) is a solution of Equation ef{eq:5.6.8} if and only if

[u'=vx=-xe^{-x}+C_1x. onumber]

Integrating this yields

[u=(x+1)e^{-x}+{C_1over2}x^2+C_2. onumber]

Therefore the general solution of Equation ef{eq:5.6.6} is

[label{eq:5.6.10} y=ue^x=x+1+{C_1over2}x^2e^x+C_2e^x.]

b. By letting (C_1=C_2=0) in Equation ef{eq:5.6.10}, we see that (y_{p_1}=x+1) is a solution of Equation ef{eq:5.6.6}. By letting (C_1=2) and (C_2=0), we see that (y_{p_2}=x+1+x^2e^x) is also a solution of Equation ef{eq:5.6.6}. Since the difference of two solutions of Equation ef{eq:5.6.6} is a solution of Equation ef{eq:5.6.7}, (y_2=y_{p_1}-y_{p_2}=x^2e^x) is a solution of Equation ef{eq:5.6.7}. Since (y_2/y_1) is nonconstant and we already know that (y_1=e^x) is a solution of Equation ef{eq:5.6.6}, Theorem 5.1.6 implies that ({e^x,x^2e^x}) is a fundamental set of solutions of Equation ef{eq:5.6.7}.

Although Equation ef{eq:5.6.10} is a correct form for the general solution of Equation ef{eq:5.6.6}, it is silly to leave the arbitrary coefficient of (x^2e^x) as (C_1/2) where (C_1) is an arbitrary constant. Moreover, it is sensible to make the subscripts of the coefficients of (y_1=e^x) and (y_2=x^2e^x) consistent with the subscripts of the functions themselves. Therefore we rewrite Equation ef{eq:5.6.10} as

[y=x+1+c_1e^x+c_2x^2e^x onumber]

by simply renaming the arbitrary constants. We’ll also do this in the next two examples, and in the answers to the exercises.

Example (PageIndex{2})

1. Find the general solution of [x^2y''+xy'-y=x^2+1, onumber] given that (y_1=x) is a solution of the complementary equation [label{eq:5.6.11} x^2y''+xy'-y=0.] As a byproduct of this result, find a fundamental set of solutions of Equation ef{eq:5.6.11}.
2. Solve the initial value problem [label{eq:5.6.12} x^2y''+xy'-y=x^2+1, quad y(1)=2,; y'(1)=-3.]

Solution

a. If (y=ux), then (y'=u'x+u) and (y''=u''x+2u'), so

[egin{aligned} x^2y''+xy'-y&=x^2(u''x+2u')+x(u'x+u)-ux &=x^3u''+3x^2u'.end{aligned}]

Therefore (y=ux) is a solution of Equation ef{eq:5.6.12} if and only if

[x^3u''+3x^2u'=x^2+1, onumber]

which is a first order equation in (u'). We rewrite it as

[label{eq:5.6.13} u''+{3over x}u'={1over x}+{1over x^3}.]

To focus on how we apply variation of parameters to this equation, we temporarily write (z=u'), so that Equation ef{eq:5.6.13} becomes

[label{eq:5.6.14} z'+{3over x}z={1over x}+{1over x^3}.]

We leave it to you to show by separation of variables that (z_1=1/x^3) is a solution of the complementary equation

[z'+{3over x}z=0 onumber]

for Equation ef{eq:5.6.14}. By variation of parameters, every solution of Equation ef{eq:5.6.14} is of the form

Since (u'=z=v/x^3), (u) is a solution of Equation ef{eq:5.6.14} if and only if

[u'={vover x^3}={1over3}+{1over x^2}+{C_1over x^3}. onumber]

Integrating this yields

[u={xover 3}-{1over x}-{C_1over2x^2}+C_2. onumber]

Therefore the general solution of Equation ef{eq:5.6.12} is

[label{eq:5.6.15} y=ux={x^2over 3}-1-{C_1over2x}+C_2x.]

Reasoning as in the solution of Example (PageIndex{1a}), we conclude that (y_1=x) and (y_2=1/x) form a fundamental set of solutions for Equation ef{eq:5.6.11}.

As we explained above, we rename the constants in Equation ef{eq:5.6.15} and rewrite it as

[label{eq:5.6.16} y={x^2over3}-1+c_1x+{c_2over x}.]

b. Differentiating Equation ef{eq:5.6.16} yields

[label{eq:5.6.17} y'={2xover 3}+c_1-{c_2over x^2}.]

Setting (x=1) in Equation ef{eq:5.6.16} and Equation ef{eq:5.6.17} and imposing the initial conditions (y(1)=2) and (y'(1)=-3) yields

[egin{aligned} c_1+c_2&= phantom{-}{8over 3} c_1-c_2&= -{11over 3}.end{aligned}]

Solving these equations yields (c_1=-1/2), (c_2=19/6). Therefore the solution of Equation ef{eq:5.6.12} is

[y={x^2over 3}-1-{xover 2}+{19over 6x}. onumber]

Using reduction of order to find the general solution of a homogeneous linear second order equation leads to a homogeneous linear first order equation in (u') that can be solved by separation of variables. The next example illustrates this.

Example (PageIndex{3})

Find the general solution and a fundamental set of solutions of

[label{eq:5.6.18} x^2y''-3xy'+3y=0,]

given that (y_1=x) is a solution.

Solution

If (y=ux) then (y'=u'x+u) and (y''=u''x+2u'), so

[egin{aligned} x^2y''-3xy'+3y&=x^2(u''x+2u')-3x(u'x+u)+3ux &=x^3u''-x^2u'.end{aligned}]

Therefore (y=ux) is a solution of Equation ef{eq:5.6.18} if and only if

[x^3u''-x^2u'=0. onumber]

Separating the variables (u') and (x) yields

[{u''over u'}={1over x}, onumber]

so

Therefore

[u={C_1over2}x^2+C_2, onumber]

so the general solution of Equation ef{eq:5.6.18} is

[y=ux={C_1over2}x^3+C_2x, onumber]

which we rewrite as

[y=c_1x+c_2x^3. onumber]

Therefore ({x,x^3}) is a fundamental set of solutions of Equation ef{eq:5.6.18}.

## 4.9: Oxidation-Reduction Reactions

The term oxidation was first used to describe reactions in which metals react with oxygen in air to produce metal oxides. When iron is exposed to air in the presence of water, for example, the iron turns to rust&mdashan iron oxide. When exposed to air, aluminum metal develops a continuous, transparent layer of aluminum oxide on its surface. In both cases, the metal acquires a positive charge by transferring electrons to the neutral oxygen atoms of an oxygen molecule. As a result, the oxygen atoms acquire a negative charge and form oxide ions (O 2&minus ). Because the metals have lost electrons to oxygen, they have been oxidized oxidation is therefore the loss of electrons. Conversely, because the oxygen atoms have gained electrons, they have been reduced, so reduction is the gain of electrons. For every oxidation, there must be an associated reduction. Therefore, these reactions are known as oxidation-reduction reactions, or "redox" reactions for short.

Any oxidation must ALWAYS be accompanied by a reduction and vice versa.

Originally, the term reduction referred to the decrease in mass observed when a metal oxide was heated with carbon monoxide, a reaction that was widely used to extract metals from their ores. When solid copper(I) oxide is heated with hydrogen, for example, its mass decreases because the formation of pure copper is accompanied by the loss of oxygen atoms as a volatile product (water vapor). The reaction is as follows:

Oxidation-reduction reactions are now defined as reactions that exhibit a change in the oxidation states of one or more elements in the reactants by a transfer of electrons, which follows the mnemonic "oxidation is loss, reduction is gain", or "oil rig". The oxidation state of each atom in a compound is the charge an atom would have if all its bonding electrons were transferred to the atom with the greater attraction for electrons. Atoms in their elemental form, such as O2 or H2, are assigned an oxidation state of zero. For example, the reaction of aluminum with oxygen to produce aluminum oxide is

Each neutral oxygen atom gains two electrons and becomes negatively charged, forming an oxide ion thus, oxygen has an oxidation state of &minus2 in the product and has been reduced. Each neutral aluminum atom loses three electrons to produce an aluminum ion with an oxidation state of +3 in the product, so aluminum has been oxidized. In the formation of Al2O3, electrons are transferred as follows (the small overset number emphasizes the oxidation state of the elements):

Equation ( ef<4.4.1>) and Equation ( ef<4.4.2>) are examples of oxidation&ndashreduction (redox) reactions. In redox reactions, there is a net transfer of electrons from one reactant to another. In any redox reaction, the total number of electrons lost must equal the total of electrons gained to preserve electrical neutrality. In Equation ( ef<4.4.3>) , for example, the total number of electrons lost by aluminum is equal to the total number gained by oxygen:

The same pattern is seen in all oxidation&ndashreduction reactions: the number of electrons lost must equal the number of electrons gained. An additional example of a redox reaction, the reaction of sodium metal with chlorine is illustrated in Figure (PageIndex<1>) .

In all oxidation&ndashreduction (redox) reactions, the number of electrons lost equals the number of electrons gained.

## Sections

The sheer size of data in the modern age is not only a challenge for computer hardware but also a main bottleneck for the performance of many machine learning algorithms. The main goal of a PCA analysis is to identify patterns in data PCA aims to detect the correlation between variables. If a strong correlation between variables exists, the attempt to reduce the dimensionality only makes sense. In a nutshell, this is what PCA is all about: Finding the directions of maximum variance in high-dimensional data and project it onto a smaller dimensional subspace while retaining most of the information.

### PCA Vs. LDA

Both Linear Discriminant Analysis (LDA) and PCA are linear transformation methods. PCA yields the directions (principal components) that maximize the variance of the data, whereas LDA also aims to find the directions that maximize the separation (or discrimination) between different classes, which can be useful in pattern classification problem (PCA “ignores” class labels).
In other words, PCA projects the entire dataset onto a different feature (sub)space, and LDA tries to determine a suitable feature (sub)space in order to distinguish between patterns that belong to different classes.

### PCA and Dimensionality Reduction

Often, the desired goal is to reduce the dimensions of a (d)-dimensional dataset by projecting it onto a ((k))-dimensional subspace (where (k<d)) in order to increase the computational efficiency while retaining most of the information. An important question is “what is the size of (k) that represents the data ‘well’?”

Later, we will compute eigenvectors (the principal components) of a dataset and collect them in a projection matrix. Each of those eigenvectors is associated with an eigenvalue which can be interpreted as the “length” or “magnitude” of the corresponding eigenvector. If some eigenvalues have a significantly larger magnitude than others, then the reduction of the dataset via PCA onto a smaller dimensional subspace by dropping the “less informative” eigenpairs is reasonable.

### A Summary of the PCA Approach

• Standardize the data.
• Obtain the Eigenvectors and Eigenvalues from the covariance matrix or correlation matrix, or perform Singular Value Decomposition.
• Sort eigenvalues in descending order and choose the (k) eigenvectors that correspond to the (k) largest eigenvalues where (k) is the number of dimensions of the new feature subspace ((k le d)).
• Construct the projection matrix (mathbf) from the selected (k) eigenvectors.
• Transform the original dataset (mathbf) via (mathbf) to obtain a (k)-dimensional feature subspace (mathbf).

## Introduction

Multifetal pregnancy reduction is defined as a first-trimester or early second-trimester procedure for reducing the total number of fetuses in a multifetal pregnancy by one or more 1. In most cases, the involved gestations will be higher-order multifetal pregnancies, defined by the presence of three or more fetuses. Throughout the document, multifetal pregnancy reduction is used to refer to reduction of a higher-order multifetal pregnancy by one or more fetuses. The special case of reduction from a twin gestation to a singleton gestation is addressed as a separate issue in the document. The ethical issues involved in multifetal pregnancy reduction are complex, and no one position reflects the variety of opinions within the membership of ACOG. The purpose of this Committee Opinion is to review the ethical considerations involved in multifetal pregnancy reduction, to analyze their role in decisions regarding multifetal pregnancy reduction, and to provide a framework that can be used by obstetrician–gynecologists in counseling patients who are considering multifetal pregnancy reduction.

## 4.9: Reduction of Order

The FCC granted a petition for a stay of the implementation of its new framework for the 4.9 GHz band requested by the Public Safety Spectrum Alliance (PSSA).

&ldquoThe Public Safety Spectrum Alliance (PSSA) would like to thank the Federal Communications Commission for its recent decision to stay its September 30, 2020 order removing 50 MHz of the 4.9 GHz band from exclusive public safety use,&rdquo the PSSA said in a statement. &ldquoIn particular, the PSSA commends Acting Chairwomen Jessica Rosenworcel for her leadership and support in helping preserve the 4.9 GHz spectrum for the public safety community. This spectrum is vital to public safety not only to help ensure nationwide interoperability, but to facilitate the introduction of new 5G capabilities into the public-safety communications ecosystem.&rdquo

Last September, the FCC approved new rules that would allow eligible states to lease some or all of the spectrum in the band to commercial or critical infrastructure entities. Before the approval of the rules, the band had been dedicated to public-safety use, but for many years, the FCC had looked at new ways to increase use of what it described as an underused band.

Many public-safety organizations, including the PSSA, came out against the proposal, arguing that the band should continue to be dedicated to public-safety use. The PSSA, in particular advocated for the FCC to give the band to the First Responder Network Authority (FirstNet Authority) to help serve first responders&rsquo future spectrum needs.

The PSSA, the Association of Public-Safety Communications Officials (APCO) International and the National Public Safety Telecommunications Coalition (NPSTC) all filed petitions for reconsideration of the new rules. Those petitions argued that the new leasing framework would undermine the availability and utility of spectrum in the band for public-safety operations. The petitions also described the rules as &ldquoarbitrary and capricious&rdquo because they lacked a basis in the record the FCC compiled for the proceeding.

While the changes to the FCC&rsquos rules became effective December 30. Under the rules, each state must designate a state lessor and that will not become effective until the commission receives approval from the Office of Management and Budget (OMB) under the Paperwork Reduction Act. After that approval, it will announce an effective date. The FCC has not yet announced an effective date for the final rule, so the leasing framework has not yet begun.

The FCC concluded that a stay of the 4.9 GHz leasing framework is appropriate.

&ldquoSpecifically, we believe that in light of the serious questions posed by the petitions for reconsideration, the possibility of irreparable harm to current and future public-safety users of the 4.9 GHz band and to our goal of facilitating greater use of this spectrum, the extent to which a stay will further the public interest, and the fact that no parties will be injured if a stay is granted, a stay is appropriate to permit the commission to address the issues raised in the petitions for reconsideration.&rdquo

Commissioner Brendan Carr released a dissenting statement to the order. In that statement, he expressed disappointment in the order.

&ldquoIn the 4.9 GHz band, it was clear that the status quo was not working,&rdquo Carr said. &ldquoAfter almost two decades, that 50 megahertz swath of spectrum remained woefully underutilized. So, we established a framework that could allow more intensive uses to flourish, including for public safety, by empowering local leaders to determine the best options for this spectrum based on their own circumstances. &hellip This is the spectrum equivalent of taking points off the board. While I am dissenting from today&rsquos decision, I remain hopeful that we can find a way to quickly put a beneficial framework back in place. And, I am open to working with my colleagues, the public-safety community and all other stakeholders on doing exactly that.&rdquo

## Resolution

• The storage location assigned to the delivery item must be included in MRP in order to reduce the PIR while posting the goods. E.g. the field 'SLoc MRP indicator' should be blank in the MRP 4 view of material master if the storage location is maintained for the material.
• The 'Consumption indicator' of the planned independent requirement (PBIM-ZUVKZ) should be the same with the 'Allocation indicator' of the requirements class of sales order and its deliveries (VBBE-VPZUO or VBBS-VPZUO), and provides the consumption (values '1', '2' or '3') in order to consume the PIR. You can check these values by double clicking the PIR in MD04 or checking the tables directly.
As to the PIR reduction while issuing the goods to customer, the table field LIPS-VPZUO for delivery can be checked, that should be the same with PBIM-ZUVKZ. Normally, the values of VPZUO in the tables VBB* and LIPS are the same.
Regarding how the the field PBIM-ZUVKZ is filled please see SAP note 772856. The value of field VPZUO for sales order/delivery is copied from the requirement class that can be customized in the transaction OVZG and its field 'Allocation ind.'. The changes made in OVZG is only applied to the sales order/delivery newly created.
Regarding how the requirement class is determined for sales order/delivery please see the note SAP 207942.
• The 'Planning consumption indicator' (PBIM-VERKZ) should provide a value which can consume the customer requirements. You can display this on the detail dialog box for the planned independent requirements in the stock/requirements list (MD04) or in the item view of the independent requirements maintenance transaction codes (MD61, MD62, MD63). Any changes made to this indicator affect the consumption immediately.

Here is the screenshot of MD04 for PBIM-ZUVKZ and PBIM-VERKZ.

SAP KBA 1715491 - Planned independent requirements consumption/reduction by sales order and delivery

Legislation has been reintroduced in Congress (HR-2337) to soften the impact of the “windfall elimination provision” that reduces Social Security benefits of those also drawing an annuity from a retirement program that does not include Social Security, including the CSRS system.

That provision reduces the Social Security benefits earned by those persons—typically before or after federal service, or as side income while federally employed—if they have fewer than 30 years of “substantial” earnings under Social Security — this year, $26,550. The maximum reduction of the Social Security benefit works out to be about$500 a month this year there is a lesser reduction for those with between 20 and 30 years of such earnings.

Under the bill, those already retired would receive an increase in benefits of the greater of $150 or the reduction they are subject to each month, starting nine months after enactment. Those retiring in 2023 or later would receive the greater of their current benefit or the benefit calculated under a new formula that sponsors say would be worth about$75 more a month to most.

Numerous bills have been offered over the years to either eliminate or soften the windfall provision, which affects some 2 million retirees, including some retirees of state and local governments. None have come close to enactment although the current measure does have the important sponsorship of the head of the House Ways and Means Committee, Rep. Richard Neal, D-Mass.

The NARFE organization said that “while this bill does not provide WEP-affected individuals the full repeal they are due, it represents a good first step in allowing some relief from this unreasonable penalty.”

The measure would not affect another Social Security reduction applying to CSRS system retirees, the “government pension offset” which reduces and in many cases eliminates a spousal or survivor Social Security benefit.

The explicit methods are those where the matrix [ a i j ] ]> is lower triangular.

### Forward Euler Edit

The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.

### Explicit midpoint method Edit

The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):

### Heun's method Edit

Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method. (Note: The "eu" is pronounced the same way as in "Euler", so "Heun" rhymes with "coin"):

### Ralston's method Edit

Ralston's method is a second-order method [1] with two stages and a minimum local error bound:

### Generic third-order method Edit

See Sanderse and Veldman (2019). [2]

### Ralston's third-order method Edit

Ralston's third-order method [3] is used in the embedded Bogacki–Shampine method.

### Classic fourth-order method Edit

The "original" Runge–Kutta method.

### Ralston's fourth-order method Edit

This fourth order method [4] has minimum truncation error.

### 3/8-rule fourth-order method Edit

This method doesn't have as much notoriety as the "classical" method, but is just as classical because it was proposed in the same paper (Kutta, 1901).

The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by

### Heun–Euler Edit

The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

The error estimate is used to control the stepsize.

### Fehlberg RK1(2) Edit

The Fehlberg method [5] has two methods of orders 1 and 2. Its extended Butcher Tableau is:

 0 1/2 1/2 1 1/256 255/256 1/512 255/256 1/512 1/256 255/256 0

The first row of b coefficients gives the second-order accurate solution, and the second row has order one.

### Bogacki–Shampine Edit

The Bogacki–Shampine method has two methods of orders 3 and 2. Its extended Butcher Tableau is:

 0 1/2 1/2 3/4 0 3/4 1 2/9 1/3 4/9 2/9 1/3 4/9 0 7/24 1/4 1/3 1/8

The first row of b coefficients gives the third-order accurate solution, and the second row has order two.

### Fehlberg Edit

The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4. Its extended Butcher Tableau is:

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

### Cash-Karp Edit

Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is

 0 1/5 1/5 3/10 3/40 9/40 3/5 3/10 −9/10 6/5 1 −11/54 5/2 −70/27 35/27 7/8 1631/55296 175/512 575/13824 44275/110592 253/4096 37/378 0 250/621 125/594 0 512/1771 2825/27648 0 18575/48384 13525/55296 277/14336 1/4

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

### Dormand–Prince Edit

The extended tableau for the Dormand–Prince method is

 0 1/5 1/5 3/10 3/40 9/40 4/5 44/45 −56/15 32/9 8/9 19372/6561 −25360/2187 64448/6561 −212/729 1 9017/3168 −355/33 46732/5247 49/176 −5103/18656 1 35/384 0 500/1113 125/192 −2187/6784 11/84 35/384 0 500/1113 125/192 −2187/6784 11/84 0 5179/57600 0 7571/16695 393/640 −92097/339200 187/2100 1/40

The first row of b coefficients gives the fifth-order accurate solution and the second row gives the fourth-order accurate solution.

### Backward Euler Edit

The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.

### Implicit midpoint Edit

The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator.

### Crank-Nicolson method Edit

The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.

### Gauss–Legendre methods Edit

These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau:

The Gauss–Legendre method of order six has Butcher tableau:

### Diagonally Implicit Runge–Kutta methods Edit

Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems. The simplest method from this class is the order 2 implicit midpoint method.

Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:

Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method:

Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:

Two-stage 2nd order Diagonally Implicit Runge–Kutta method:

Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:

Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:

Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:

Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method

### Lobatto methods Edit

There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.

#### Lobatto IIIA methods Edit

The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule:

The fourth-order method is given by

This methods are A-stable, but not L-stable and B-stable.

#### Lobatto IIIB methods Edit

The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by

The fourth-order method is given by

Lobatto IIIB methods are A-stable, but not L-stable and B-stable.

#### Lobatto IIIC methods Edit

The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by

The fourth-order method is given by

They are L-stable. They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.

#### Lobatto IIIC* methods Edit

The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature. [6] The second-order method is given by

Butcher's three-stage, fourth-order method is given by

These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for s = 2 is sometimes called the explicit trapezoidal rule.

#### Generalized Lobatto methods Edit

One can consider a very general family of methods with three real parameters ( α A , α B , α C ) ,alpha _)> by considering Lobatto coefficients of the form

For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by

Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s − 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. The first order Radau method is similar to backward Euler method.

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A percent off of a product or service is a common discount format. A percent off of a product means that the price of the product is reduced by that percent. For example, given a product that costs $279, 20% off of that product would mean subtracting 20% of the original price, from the original price. For example: 20% of$279 = 0.20 × 279 = $55.80 You would therefore be saving$55.80 on the purchase for a final price of $223.20. For this calculator a "stackable additional discount" means getting a further percent off of a product, after a discount is applied. Using the same example, assume that the 20% discount is a discount applied by the store to the product. If you have a coupon for another 15% off, the 15% off would then be applied to the discounted price of$223.20. It is not a total of 35% off of the original price, it is less:

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