text:
how can i watch black wall street burning
Discover what to watch this November, including a documentary series that dives deep into the Marvel Universe, a reboot of a beloved '90s animated series, and a special that celebrates a very important Star Wars holiday. His books include Black Wall Street, Up From the Ashes, Acres of Aspiration, Apartheid in Indian Country, and Tulsa's Historic Greenwood District, Available for everyone, funded by readers. Even now to this day the burning of Greenwood STILL goes unpaid. GREAT MOVIE.... They gunned people down and set fires throughout the district. The Tulsa Lynching of 1921: A Hidden Story, State of Fear: Murder and Memory on Black Wall Street. White employers made arrangements to grant the African American citizens their freedom by issuing wearable green tags which, in order to be valid, needed to be signed by a white employer. While the precise source of this lapse is unclear, it is crystal clear that the decision poses a significant threat to the physical and emotional wellbeing of many Tulsans of all stripes. Has it All Today Americans still debate what occurred between Dick Rowland and Sarah Page in the Drexel building elevator on downtown of Tulsa's Main Street on May 30, 1921. This realization led to the city beginning a reconstruction commission. By the 1920s, the African American community in Tulsa, Oklahoma, emerged as a nationally renowned entrepreneurial center. They would also try to destroy every vestige of black prosperity."9 The Tulsa Race Race Riot was the first race riot of its kind in America. Drama. It's absolutely absurd that people try to not only avoid responsibility but also try to erase it as a whole. | So do enduring inequalities: disparities in education, healthcare, criminal justice, employment and even lifespan remain a fact of life for black Tulsans. Historians have dubbed this blood-soaked era the "nadir" of race relations in America. All rights reserved. Some African Americans fled Tulsa, never to return. While Black Wall Street made a strong comeback following World War II, it slowly waned in the decades to follow thanks to urban renewal initiatives that strained its economic stability. Post-civil war land allotments awarded to black members of Native American tribes helped fuel the economic engines of black communities throughout Oklahoma, including the Greenwood district. The Tulsa race massacre of 1921, when mobs of white residents rampaged. Planes, likely privately owned, dropped incendiary devices on the Greenwood district. It is interesting to read about this because they're still, maybe not surprisingly, are racial riots happening in the US right now. Integration undermined the financial foundation of Black Wall Street's largely insular black economy. In lynching them, white vigilantes were exacting lethal retribution for the crime of being black men with more business acumen than a white competitor. That thing he caught happened to be the arm of a white elevator operator, Sarah Page.1. On top of that, the city refused to accept responsibility for the riot or for the reconstruction and blamed the mayhem on the African American residents of Greenwood.14, The white political and business leaders of Tulsa were ashamed once they recognized the reality of the unapologetic racism displayed by the white mobs during the night of which the destruction of the primarily African American portion of the city ensued. Of all possible times and venues to launch a re-election bid, this choice reeks of insensitivity – poor judgment at best, calculated callousness at worst. : The Story of Larry Nassar and his 499 Victims, "You have the right to an attorney..." or Not, Live Rattlesnakes and Rock n' Roll: The Story of ZZ Top's "Worldwide Texas Tour", Impaled: The Impossible Story of Phineas Gage, How the Soviet Union Compelled the United States to Build the World's Fastest Jet: The Lockheed A-12 OXCART, Jonas Salk: Discovering The Shot that Saved the World - 20th century Polio Virus Epidemic, American Enemy? (co-director). And while the officers conducted the investigation quietly, The Tulsa Tribune printed their article, intentionally portraying the incident with a skewed perspective.17. The Tulsa race massacre… The violence and vandalism resulted in upwards of 300 people killed, with most being African American. Watch Black Wall Street Burning Online | Vimeo On Demand This film is a retelling of the worst act of American terrorism and racism in American History. Even when you google the riot and read about it through Wikipedia it states that there were only 36 deaths (according to 1921). OW Gurley owned rental properties, a rooming house, a grocery store, and a hotel. In less than 24 hours, a white mob reduced a vibrant, 35-block area to rubble and dead bodies. Renouncing Citizenship for Freedom and Enhancing Rights for All Americans, Henry Lee Lucas: The Tellings of a Serial Confessor, The Non-Stick Chemical That Stuck DuPont with a Stiff Bill In The Courtroom, Hope and Grief: Robert Kennedy's Speech on the night of Dr. King's Assassination, Katherine Johnson, The Brains of Freedom 7, Adrian Vidal: A Tejano Caught Between Two Wars, The World's Amazing (Accidental) Discovery of Penicillin, The 1917 Bath Riots: An Untold Story Of The Border, The Warren Commission Report: Conspiracy Theories Addressed on the Assassination of President John F. Kennedy, US-Emergence of Modern America (1890-1930), US-Postwar United States (1945-early 1970s), US-Contemporary United States (1968-present), Richard Lloyd Jones, "Nab Negro for Attacking Girl in Elevator,", Scott Ellsworth, "Tulsa's Successful History,", Griff Palmer, "Tulsa's Greenwood Centre Was Once 'Black Wall Street of the Southwest',", Vincent E. Miles, "Tulsa Oklahoma, Race Riot of 1921,", Alfred L. Brophy, "Tulsa, Oklahoma, Race Riot of 1921,", Jim Yardley, "Panel Recommends Reparations In Long-Ignored Tulsa Race Riot,". A Student Organization of St. Mary's University of San Antonio Texas, College of Arts, Humanities, and Social Sciences, St. Mary's University, Department of History, St. Mary's University, Department of Political Science, St. Mary's University, Center for Catholic Studies, St. Mary's University, The Learning Assistance Center, St. Mary's University, Louis J. Blume Library Services, St. Mary's University, STRIVE Career Center, St. Mary's University, Academic Technology Services, St. Mary's University, US-Colonization and Settlement (1585-1763), US-Revolution and the New Nation (1754-1820s), US-Civil War & Reconstruction (1850-1877), Gun Violence in America: The Sandy Hook Story, E-Cigarettes: An Incredible Invention or Complete Catastrophy? At this time, the Tulsa Tribune was a Republican paper, edited and published since 1919 by the controversial racist Richard Lloyd Jones. And, of course, systemic racism continued unchecked. Adam Hergenreder Shares His Catastrophic Experience, Drugs, Alcohol, and Arrests: The Downside to Robert Downey Jr.'s Acting Career, Deaths, Deformities, and Illnesses: The Consequences of Agent Orange during the Vietnam War, Raping and Killing: "Casualties of War" in Vietnam, Breaking Silence or Just Speaking Louder? Also, if these type of stories are not brought into history books, how can we learn? How sad that they were attempting to cover up this great injustice after they caused so many African Americans to be injured and lose their property. They planned to use the land as an industrial park or a railroad station, and passed strict building ordinances requiring the use of fireproof material.15 The cost of the requirements made the rebuilding of Greenwood nearly impossible for its African American residents. Glad that the author chose to tell this story though because those people to this dy have not gotten the justice they deserve. The area, centered around Tulsa's Greenwood district, was known as "Black Wall Street". On top of those stipulations, African American victims of the riot were expected to rebuild their community from scratch which inadvertently forced many to live in tent-cities. In Tulsa, "Black Wall Street Burning" will show Feb. 14-20 at Circle Cinema, 10 S Lewis Ave. To mark Black History Month, Riggins and Brown will participate in a Q&A following the 7:30 p.m. Feb. 15 screening at Circle Cinema. Property damage, conservatively estimated, ranged from $1.5m to $2m, well over $25m in present value. What upset me the most was that the town tried to repurpose the land rather than restore it to the way it was prior to the riot. In recent years forensic scientists have been searching Tulsa for possible mass graves. As witnesses later recounted the events of that night, they argued that the white rioters had forgotten about Dick Rowland entirely, and their motivation became to "settle for nothing less than scorched earth. So do enduring inequalities. This article brings back a lot of unsettling memories of America's racist past, and its very sad to hear such an innocent incident transform into a community wide riot, ultimately effecting lives and homes of African american citizens.
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INVESTIGATION: What's causing colleges to falter financially?
At least five Dayton area colleges have struggled financially over the last few years.
By Max Filby
While higher education is facing leaner times throughout the country, area universities have faced tough finances for a variety of reasons, including some that were self inflicted.
At least five colleges in the region have struggled financially over the last few years. A Dayton Daily News investigation this week looks at these area colleges and others to see what is causing the financial issues.
Some experts told us there are too many colleges and universities in Ohio and one higher education analyst suggested that more than 500 schools would close across the United States in the next decade.
Antioch College in Yellow Springs has struggled to remain open in the past decade and Urbana University in Champaign County had to be purchased by another college in order to survive. The area's two historically black schools, Central State University and Wilberforce University, have also struggled with finances over the last few years.
RELATED: Sinclair Community College could raise tuition, give employees pay raises
And Wright State University is in the middle of dealing with a financial crisis that has included millions in budget cuts.
Here's what happened at each of the five colleges.
Wright State recently implemented $30.8 million of budget cuts.
The move was an attempt to correct years of overspending as WSU has spent more money than it's generated since 2012.
RELATED: WSU trustees approve more than $30.8 million in budget cuts
The school announced earlier this month that it would lay off around 57 employees as part of an overall elimination of 189 positions, which will save the university more than $13.9 million.
The school could also eliminate its men's and women's swimming and diving teams, on top of other cuts.
Officials have said the university is heading for state fiscal watch and will likely be placed on it by 2019. Despite recent cuts, Wright State's financial problems have not been fully resolved.
The school still has to rebuild its reserves by adding around $45 million over the next few years, state and WSU officials have said. To boost reserves, WSU will look to up enrollment, sell assets and as a last resort, make more cuts.
Antioch College's residential campus closed in 2008 in the wake of significant declining enrollment and financial challenges.
The 2008 closure came after the college had expanded to several branch campuses across the country, known as the Antioch University system. Those branches remain open and there is currently a branch in Yellow Springs called Antioch University.
RELATED: Antioch University in Yellow Springs could move to south Dayton
In 2009, a group of Antioch College alumni formed a group to reopen the college as a separate entity. They did so in 2011.
While Antioch College was seemingly brought back from the dead, it still faces the prospect of rebuilding its enrollment and becoming competitive enough to prevent what happened once before from happening again. The college's alumni have been a big force in keeping the school afloat, donating their time and hundreds of thousands of dollars.
Urbana University
In 2014, Urbana University officials announced that the school would be sold to Franklin University in Columbus.
Financial details about the agreement were never disclosed. But, the purchase was necessary because of long-term debt at Urbana and administrative decisions that led to deficit spending, according to information from the universities and tax forms filed with the Internal Revenue Service.
RELATED: Urbana University begins first year after rare, rapid acquisition
The financial issues also led Urbana to lay off nine faculty and staff in 2014 as well.
Urbana now functions as something of a branch campus of Franklin University, though it has maintained its sports teams and name. Later this year, that designation is expected to be made official once final approval is granted by the Higher Learning Commission, said David Decker, president of Franklin University.
Central State University
Central State University was placed on state fiscal watch in 2015 and came off of it this past April.
CSU was placed on fiscal watch after its finances dropped below a state threshold used to gauge the financial health of Ohio's public colleges.
RELATED: Central State removed from fiscal watch as Wright State edges toward it
Declining student enrollment and students' difficulty in qualifying for federal financial aid were then major challenges for the historically-black university in Wilberforce, president Cynthia Jackson-Hammond has said.
CSU's fall enrollment decreased by more than 15 percent from 2,068 in 2013 to 1,751 in 2015, according to the state. As of fall 2016, that number had ticked up to 1,804, according to a preliminary state enrollment report.
Central State has faced a number of other problems specific to it being an HBCU. Jackson-Hammond traveled to Washington, D.C. in February to meet with President Donald Trump, where leaders of black colleges asked for more funding to update and better sustain their institutions.
Wilberforce University
Like CSU's leader, Wilberforce University president Herman Felton also traveled to D.C. to seek more support for historically black colleges.
Wilberforce University has faced its share of financial difficulties as well.
RELATED: Wilberforce University announces layoffs, pay cuts and furlough for employees
The university cut $750,000 from its payroll budget in November. Layoffs, furloughs and pay cuts were implemented in May as Wilberforce continues to right-size its budget.
Wilberforce University, the oldest historically black private university in the country, was at risk of losing its accreditation from mid-2014 through most of 2015, due to declining enrollment. The school was issued a "show cause" order from the Higher Learning Commission that was later lifted in November 2015 after enrollment increased by more than 85 percent to around 650 students.
Read the Dayton Daily News' full investigation online Thursday and in print on Sunday.
Fairborn Schools extends remote learning through end of week
New list: Which schools are remote or closed, for how long?
Humane society honors legend with #BettyWhiteChallenge
Community helps homeless Dayton students, who will receive hygiene...
Max Filby

redpajama_set_name: RedPajamaCommonCrawl


text:
BRIDGET MOYNAHAN, DANE DEHAAN AND ANNA WOOD TO HOST DUNKIN' CELEBRITY MINI GOLF TOURNAMENT AT THE 2019 TRAVELERS CHAMPIONSHIP
Home / BRIDGET MOYNAHAN, DANE DEHAAN AND ANNA WOOD TO HOST DUNKIN' CELEBRITY MINI GOLF TOURNAMENT AT THE 2019 TRAVELERS CHAMPIONSHIP
HARTFORD, Conn., June 14, 2019 — The Travelers Championship today announced that actors Bridget Moynahan, Dane DeHaan and Anna Wood will host this year's Dunkin' Celebrity Mini Golf Tournament. The event will take place in the Stanley Black & Decker Fan Zone Wednesday, June 19 at TPC River Highlands.
Shotgun starts at 9:30 and 10:30 a.m. will feature three-player teams. The two winning teams each will receive $2,500 to donate to the charities of their choice. For the seventh consecutive year, Renee DiNino from iHeartMedia Communities will serve as master of ceremonies for the event.
"The Celebrity Mini Golf Tournament is one more way for us to get more people involved and generate more money for charity," said Travelers Championship Tournament Director Nathan Grube. "We are grateful for everyone's participation – the Hollywood stars, the Connecticut news teams, the local sports team mascots – they all play an important part in making the day a success."
Bridget Moynahan currently stars in the CBS drama "Blue Bloods." Her other television credits include "Sex and the City" and J.J. Abrams's "Six Degrees." Moynahan's breakthrough role was in the film "Coyote Ugly." She went on to appear in "Serendipity," "The Sum of All Fears," "I, Robot" and "Lord of War." Most recently, she has appeared in "John Wick," "John Wick 2" and "Drunk Parents." Moynahan also serves on the board of directors of the Travelers Championship's primary charitable beneficiary, The Hole in the Wall Gang Camp.
Dane DeHaan is one of Hollywood's most sought-after actors, best known for his roles in the movies "Chronicle," "The Amazing Spider-Man 2," "Kill Your Darlings," "Life" and "Valerian and the City of a Thousand Planets." DeHaan can currently be seen in "The Kid," in which he portrays Billy the Kid.
Anna Wood is currently starring in the CBS drama "The Code." Her other television credits include "Royal Pains," "The Good Wife," "NCIS: Los Angeles," "Cold Case," "Madam Secretary," "Mad Men," "Brothers & Sisters" and "The Following." On the big screen, Wood had a key supporting role in the super-powered feature film "Chronicle."
"Dunkin' is proud to sponsor the Celebrity Mini Golf Tournament at the Travelers Championship," said Molly Loh, Dunkin' Field Marketing Manager for the Hartford market. "Helping the community is a key part of our values here at Dunkin', and we're happy to partner with a tournament that shares those values."
Moynahan, DeHaan and Wood are longtime friends of The Hole in the Wall Gang Camp and will be playing mini golf with campers during the Dunkin' Celebrity Mini Golf Tournament.
Celebrity mini golf is just one of the exciting activities in the Stanley Black & Decker Fan Zone during the week. There, fans can enjoy the Liberty Bank Kids Zone with arts and crafts, LEGO® activities and Bubba Watson's Sweet Spot candy store, try out a swing simulator, play on the Creative Dimensions mini golf course and more.
The Travelers Championship will take place June 17–23 at TPC River Highlands. Tickets are on sale at www.TravelersChampionship.com.
About the Travelers Championship
The Travelers Championship, held in Cromwell, Connecticut, is one of the premier events on the PGA TOUR and part of the FedExCup series. Named by the PGA TOUR in 2017 and 2018 as the recipient of the "Players Choice" award, the Travelers Championship proudly supports the TOUR's Tradition of Giving Back by donating 100 percent of net proceeds to charities. Travelers, a leading provider of property and casualty insurance for auto, home and business, is the Official Property Casualty Insurance Provider of the PGA TOUR. The company became the tournament's title sponsor in 2007. Travelers has been doing business in the Hartford community for more than 160 years and has been a corporate sponsor of this event each year since its inception in 1952. Complete details are available at www.TravelersChampionship.com.
Tara Gerber, Travelers Championship, 860.502.6815, tgerber@travelerschampionship.com


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Q: Fastest way to query data by a specific date and data by next older date from oracle table? I am developing a web-application using c# and MVC in .net environment. The Database is Oracle and connected via the DevArt Connector to the Entity Framework from which I created my entities Database first.
The Table that gets queried has some Data columns and a Date Column.
The specific problem is: The User will enter a specific date for the query and I will get a List of Elements (~10k) that contain this specific date and get their belonging data. Additionally I need the data from EVERY ELEMENT IN THE LIST for THE NEXT OLDER DATE.
Note: There will be Datasets with Dates for these elements that are younger or older than the date the user enters and sometimes there is no younger date, so max() or Top2 won't work just so easily.
This is what I came up with:
var query = context.BsgApo
                    .Where(
                        a => ListOfPznsToDate.Contains(a.Pzns)
                            && a.Date <= UserDate
                                && context.BsgApo.Any(
                            sub =>
                                sub.Pzns == a.Pzns &&
                                sub.Date < a.Date)
                                    &&
                                context.BsgApo.Any(
                                    negSub => negSub.Pzns == a.Pzns &&
                                        negSub.Date > a.Date &&
                                        negSub.Date < UserDate
                                        ) == false
                                )
                    .OrderBy(eL => eL.Pzns).ThenBy(eL => eL.Date)
                    .Select(MapBsgApoToPreisaenderungsErgebnisListe)
                    .ToList();

First, I query the table by a list of elements where each element has a unique identifier (pzn) so I will get just the elements I am interested in AND that have a date that is equal or younger than the one the user entered.
Then I do a sub-query because I want the dataset of the (maybe existent) smaller date. If there is one, then I check if there is none between the smaller one and the date the user entered.
This actually works just halfway, I miss some data and the query takes some seconds.
So how can I add the data from the next older date to my list of elements in a correct and fast way?

A: You should group the data by pzn and within each group take the two most recent entries. Also filter the groups that have > 1 entries:
var query = context.BsgApo
                   .Where(a => a.Date <= UserDate)
                   .GroupBy(eL => eL.Pzns)
                   .Where(grp => grp.Count() > 1)
                   .SelectMany(grp => grp.OrderByDescending(eL => eL.Date).Take(2))
                   .OrderBy(eL => eL.Pzns).ThenBy(eL => eL.Date)
                   .Select(MapBsgApoToPreisaenderungsErgebnisListe)
                   .ToList();

(leaving out some pieces that are not relevant here)

redpajama_set_name: RedPajamaStackExchange


Gillard, S. (2000). Psychological 'Conflict Mapping' in Bosnia & Hercegovina: Case Study, Critique and the Renegotiation Theory. Bradford, Centre for Conflict Resolution, Department of Peace Studies, University of Bradford. CCR Working Papers: No. 6.


text:
Brother vs. Brother: Find out if William or Harry won the polo match
Princess Diana's sons competed against each other at a charity game
July 10, 2019	 by Alexandra Hurtado
Big brother Prince William was crowned the winner of today's polo match. The Duke of Cambridge's team beat his younger brother Prince Harry's at the King Power Royal Charity Polo Day for the Khun Vichai Srivaddhanaprabha Memorial Polo Trophy on Wednesday, July 10. The royal siblings competed in the charity match, which raised more than £1 million for 15 charities supported by William and Harry, including African conservation organizations, rugby charities and the Invictus Games Foundation.
Harry (in red) and William (in blue) faced off against each other in a charity polo match on July 10
The polo match was held in honor of Leicester City chairman Vichai Srivaddhanaprabha, who was an avid polo supporter and president of Ham Polo Club for four years. Vichai was killed in a helicopter crash at the King Power football stadium in Leicester last October. Princess Diana's sons appeared to be in great spirits as they faced off against each other.
GALLERY: ALL THE BEST PHOTOS FROM LOUIS AND ARCHIE'S POLO PLAYDATE
Harry, 34, and William, 37, had special cheerleaders on the sidelines as they competed in the game, their respective wives and children. Prince Louis and Archie Harrison made their polo match debut, joining moms Kate Middleton and Meghan Markle, in addition to Prince George and Princess Charlotte.
The royal brothers had the support of their wives and children at the game
The match marked Archie's first public appearance, as well as the first time the Cambridge and Sussex cousins have been pictured publicly together. Prince Louis showed off his cheeky side at the royal affair playing with his mother's sunglasses and running around the field. Meanwhile, George and Charlotte played with a soccer ball and polo mallets. Archie, on the other hand, spent the game cradled in his doting mother's arms. The newborn's attendance at the sporting event comes days after his private christening at Windsor Castle, which took place exactly two months after his birth.
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What do Prince Harry and Meghan have in common with Cardi B?
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text:
\section{Introduction}

In this paper, we consider the complex continuous-time algebraic
Riccati matrix equation (CARE) defined by
\begin{align}
A^*X+XA-XKX+Q=0,
\label{eq:CARE}
\end{align}
where $A,~K,~Q\in\mathbb{C}^{n\times n}$,
$K=K^*$, $Q=Q^*$, and $X$ is an unknown matrix.

The CARE arises in many areas such as linear and non-linear optimal control systems \cite{P.L.1995Algebraic}-\cite{G.1990Worst},
 Wiener-Hopf factorization of Markov chains \cite{D.1982potential},
 total least squares problems \cite{B.J.1992total}, spectral factorization of rational matrix functions \cite{K.I.1981factorization}-\cite{I.M.1987inverse},
 matrix sign functions \cite{J.1980linear}-\cite{A.1991invariant}, and transport theory \cite{J.W.1999nonsymmetric}. For example, consider the linear time-invariant system represented by
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)+B\bm{u}(t),~~\bm{p}(0)=\bm{p}_0,
\end{align*}
where $B\in \mathbb{C}^{n\times m}$, $A$ is defined by \eqref{eq:CARE},
$\bm{p}(t)\in\mathbb{C}^{n}$ and $\bm{u}(t)\in\mathbb{C}^{m}$ are the state and control vectors of the system.
We can represent $u(t)=-R^{-1}B^{*}\bm{p}(t)$ for a vector $\bm{x}(t)$ such that the cost functional
\begin{align}
J(u)=\int^{t_1}_{t_0} [\bm{p}^{*}(t)Q \bm{p}(t)+\bm{u}^*(t)R\bm{u}(t)]\,dt+\bm{p}^{*}(t_1)Z_1 \bm{p}(t_1)
\label{eq:def-cost-J}
\end{align}
is minimized, where the cost penalty matrix $Q$ defined by \eqref{eq:CARE} is positive semi-definite, and
the cost penalty matrices $R,~Z_1$ are positive definite.
Taken $K=BR^{-1}B^*$, defined by \eqref{eq:CARE}, the state equation becomes
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)-K\bm{x}(t).
\end{align*}
The cost function reduces to
\begin{align*}
\hat{J}(u)=\int^{t_1}_{t_0} [\bm{p}^{*}(t)Q \bm{p}(t)+\bm{x}^*(t)K\bm{x}(t)]\,dt+\bm{p}^{*}(t_1)Z_1 \bm{p}(t_1).
\end{align*}
We wish to choose $\bm{x}(t)$ to minimize $\hat{J}$.
This can be accomplished by recognizing the fact input
$\bm{x}(t)+\delta\hat{\bm{x}}(t)$,
which is some deviation from the desired stabilizing input.
The corresponding state is $\bm{p}(t)+\delta\hat{\bm{p}}(t)$ (by linearity).
It can be shown that $J$ is a minimum when the derivative of the adjusted
cost function $\hat{J}$ with respect to $\delta$ is zero \cite{J.2008Riccati}, \textit{i.e.,}
\begin{align*}
\int^{t_1}_{t_0} [\bm{p}^{*}(t)Q \hat{\bm{p}}(t)+\bm{x}^*(t)A\hat{\bm{p}}(t)-\bm{x}^*(t)\dot{\hat{\bm{p}}}(t)]\,dt
+\bm{p}^{*}(t_1)Z_1 \hat{\bm{p}}(t_1)=0.
\end{align*}
Using integration by parts and $\hat{\bm{p}}(t_0)=0$, we have
\begin{align*}
\int^{t_1}_{t_0} [\bm{p}^{*}(t)Q+\bm{x}^*(t)A+\dot{\bm{x}}^*(t)]\hat{\bm{p}}(t)\,dt
+[\bm{p}^{*}(t_1)Z_1-\bm{x}^*(t_1)] \hat{\bm{p}}(t_1)=0.
\end{align*}
If the system is controllable, then a suitable choice of $\hat{\bm{x}}(t)$
gives any $\hat{\bm{p}}(t)$. This leads to the
following requirements
\begin{align*}
\begin{cases}
\bm{p}^{*}(t)Q+\bm{x}^*(t)A+\dot{\bm{x}}^*(t)=0,\\
\bm{p}^{*}(t_1)Z_1-\bm{x}^*(t_1)=0.
\end{cases}
\end{align*}
Now the system becomes
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)-K\bm{x}(t).
\end{align*}
Since there exists a state transition matrix by linearity
\begin{align*}
X(t,t_1)=
\begin{pmatrix}
X_{11}(t,t_1) & X_{12}(t,t_1)\\
X_{21}(t,t_1) & X_{22}(t,t_1)
\end{pmatrix}
\end{align*}
such that
\begin{align*}
\begin{pmatrix}
\bm{p}(t)\\
\bm{x}(t)
\end{pmatrix}
=
\begin{pmatrix}
X_{11}(t,t_1) & X_{12}(t,t_1)\\
X_{21}(t,t_1) & X_{22}(t,t_1)
\end{pmatrix}
\begin{pmatrix}
\bm{p}(t_1)\\
\bm{x}(t_1)
\end{pmatrix}.
\end{align*}
Then, we have $\bm{x}(t)=X(t)\bm{p}(t)$, where $X(t)$ is the solution
of the continuous time differential Riccati equation \cite{J.2008Riccati}
\begin{align}
-\frac{dX(t)}{dt}=A^*X(t)+X(t)A-X(t)KX(t)+Q.
\label{eq:dCARE}
\end{align}
When $t_1\rightarrow \infty$, we get a steady state stabilizing solution $\dot{X}(t)=0$.
The continuous time differential
Riccati equation \eqref{eq:dCARE} reduces to the CARE \eqref{eq:CARE}. The optimal choice of input that minimizes the infinite horizon cost function
\begin{align*}
J(u)=\int^{\infty}_{t_0} [\bm{p}^{*}(t)Q \bm{p}(t)+\bm{u}^*(t)R\bm{u}(t)]\,dt
\end{align*}
is $\bm{u}(t)=-R^{-1}B^*X\bm{p}(t)$.

Therefore, there are many scholars pay much attention to studying the  CARE \eqref{eq:CARE}. Many works are concerned with the existence of
positive (semi-) definite solution for this equation.
In 1961, Kalman derived the existence condition
of the positive definite solution using observability
and controllability \cite{R.1961new}. Subsequently, the existence conditions of the positive (semi-) definite solution have been
considerably investigated under suitable assumptions in \cite{W.1968matrix}-\cite{D.B.B.2012stabilizing}.

\begin{thm}
\cite{D.B.B.2012stabilizing}
For the CARE \eqref{eq:CARE}, if the pair $(A,K)$ is stabilizable
and $(A,Q)$ is detectable, then this equation has a unique Hermitian positive semi-definite
solution $X$. $(A,K)$ is called stabilizable if there exists a matrix
$F\in\mathbb{C}^{n\times n}$ such that $A-KF$ is stable, \textit{i.e.,}
all eigenvalues are in the open left half-plane $\mathbb{C}_{-}$. The pair $(A,Q)$ is called detectable if $(A^*,Q^*)$ is stabilizable.
\end{thm}

Additionally, there are many works focusing on the numerical algorithms for the CARE \eqref{eq:CARE}, such as
the Schur method \cite{R.B.D.2008parallel},
the matrix sign function \cite{R.1987solving},
the structure-preserving doubling algorithm \cite{E.H.W.W.2004structure},
and Krylov subspace projection method \cite{B.S.2013numerical}-\cite{B.H.J.2016preconditioned}. Kleinman \cite{D.1968iterative}, Banks and Ito \cite{H.K.1991numerical} applied the Newton method, due to its quadratic convergence, to solve this equation.
Nevertheless, at each Newton iteration step, a Lyapunov equation needs to be solved to get next iteration solution.
Therefore, Navasca and Morris in \cite{K.C.2005solution}-\cite{K.C.2006iterative}
combined the Newton method with a modified alternating-direction implicit (ADI) method.
Benner and his coauthors used the variant of Newton-ADI algorithm to
solve large-scale Riccati equations \cite{P.H.J.2008parameter}-\cite{P.J.2010Galerkin}.
Moreover, Feitzinger et al. in \cite{F.T.E.2009inexact}-\cite{P.M.2016inexact} proposed
and analyzed the inexact Newton-ADI method. Recently, \cite{M.A.2021Hermitian} has proposed an iteration
 scheme and given its theoretical analysis. It should be noticed that this algorithm still needs to solve a Lyapunov equation twice at each iteration step. Based on this, we concentrate on solving the CARE \eqref{eq:CARE} by combing the Newton method
with the generalized alternating directions implicit (GADI).

 The remaining part of this paper is organized as follows.
In Section \ref{sec:Newon-GADI}, we propose two Newton-GADI methods, including the Newton-GADI
and inexact Newton-GADI algorithms. Moreover, we show the convergence analysis of the Newton-GADI.  A practical method is given to select the quasi-optimal parameter.
In Section \ref{sec:NE}, some numerical examples are devoted to showing the effectiveness of
the proposed algorithms.  We draw some conclusion and remarks in the last section.

Throughout this paper, let $\mathbb{C}^{n\times m}$ be the set of all $n\times m$ complex matrices. $I_n$ is the identity matrix of order $n$.
If $A\in \mathbb{C}^{n\times n}$, the symbols $A^{\ast}$, $A^{-1}$ and $\|A\|_{2}$ express the conjugate transpose,
the inverse, the spectral norm of $A$, respectively. The eigenvalue and singular value sets of $A$ are denoted as $\Lambda(A)=\{\lambda_{i}(A),~i=1, 2,\cdots, n\},
~~\Sigma(A)=\{\sigma_{i}(A),~i=1, 2,\cdots, n\},$ where $\lambda_{i}(A)$ and $\sigma_i(A)$ represent
the $i$-th component, arranged in non-increasing order.  $\rho(A)=\max\limits_{1\leq i\leq n}\{|\lambda_{i}(A)|\}$
represents the spectral radius of $A$.

\section{Newton-GADI algorithms}
\label{sec:Newon-GADI}

It is well known that one classical approach to solving the  CARE \eqref{eq:CARE} is to tackle its nonlinearity
with a Newton-type method. In this section, we apply the Newton method to transform this equation into a Lyapunov equation.  Then we use the GADI method to solve the Lyapunov equation. The whole framework is denoted by the Newton-GADI.
To save computational amount, the inexact Newton-GADI method is presented. Further, we give the convergence analysis of the Newton-GADI algorithm.

\subsection{Iteration schemes}

Now we introduce the usual scheme of the Newton method
for solving the CARE \eqref{eq:CARE}.

Define the mapping $\mathcal{R}: \mathbb{C}^{n\times n}\rightarrow \mathbb{C}^{n\times n}$:
\begin{align*}
\mathcal{R}(X)=A^*X+XA-XKX+Q,~X\in \mathbb{C}^{n\times n},
\end{align*}
where $A,~K,~Q$ are defined in \eqref{eq:CARE}. The first Fr$\acute{\mbox{e}}$chet derivative of $\mathcal{R}$
at a matrix $X$ is a linear map
$\mathcal{R}'_X: \mathbb{R}^{n\times n}\rightarrow \mathbb{R}^{n\times n}$ given by
\begin{align*}
\mathcal{R}'_X{(E)}=(A-KX)^*E+E(A-KX).
\end{align*}
The Newton method for the CARE \eqref{eq:CARE} is
\begin{align}
X_{k+1}=X_{k}-(\mathcal{R}'_{X_k})^{-1}\mathcal{R}(X_k),~k=0,1,\cdots,
\label{eq:Newton-scheme}
\end{align}
given that the map $\mathcal{R}'_{X_k}$ is invertible.
The Newton iteration scheme \eqref{eq:Newton-scheme} is equivalent to
\begin{align}
(A-KX_k)^*X_{k+1}+X_{k+1}(A-KX_k)+X_k K X_k+Q=0.
\label{eq:Newton-CARE}
\end{align}
The convergence property for the Newton scheme \eqref{eq:Newton-CARE} is illustrated by the following result.

\begin{thm}
\cite{D.1968iterative}
For the CARE \eqref{eq:CARE}, assume that $(A,K)$ is stabilizable and $(A,Q)$ is detectable,
starting with any Hermitian matrix $X_0$ such that $A-KX_0$ is stable,
the sequence of Hermitian matrices $\{X_{k}\}_{k=1}^{\infty}$
determined by \eqref{eq:Newton-CARE}
quadratic converges to the Hermitian
positive semi-definite solution $X_*$ of \eqref{eq:CARE}.
In other words, there exists a constant $\delta>0$ such that
\begin{align*}
\|X_k-X_*\|\leq \delta \|X_{k-1}-X_*\|^2,~~k=0,1,\cdots,
\end{align*}
where $\|\cdot\|$ is any given matrix norm.
Moreover, the iteration sequence  $\{X_{k}\}_{k=1}^{\infty}$ has a monotone convergence behavior, \textit{i.e.,}
\begin{align*}
0\leq X_*\leq \cdots \leq X_{k+1}\leq X_{k}\leq \cdots \leq X_1.
\end{align*}
\end{thm}

Denote $A_k=KX_k-A$ and $F(X_k)=X_k K X_k+Q$, where the mapping $F: \mathbb{C}^{n\times n}\rightarrow \mathbb{C}^{n\times n}$.
Actually, due to $A-KX_k$ is stable, then $A_k$ is stable,
\textit{i.e.,} $\mbox{Re}(\lambda_i(A_k))>0$.
The iteration scheme can be written as
\begin{align}
A_k^*X_{k+1}+X_{k+1}A_k=F(X_k).
\label{eq:Newton-Lyapunov}
\end{align}
It is required to solve the Lyapunov equation \eqref{eq:Newton-Lyapunov}
in each Newton iteration step.
We recall an existence and uniqueness
theorem for this equation.
\begin{thm}
The Lyapunov equation
\begin{align*}
\hat{A}^*Y+Y\hat{A}+Z=0
\end{align*}
has a uniquely Hermitian positive semi-definite solution $Y\in\mathbb{C}^{n\times n}$ if and only if $\hat{A}\in\mathbb{C}^{n\times n}$ is stable, for any Hermitian matrix $Z\in\mathbb{C}^{n\times n}$.

\end{thm}
We apply the GADI framework in \cite{K.X.J.2021general} to solve the Lyapunov equation \eqref{eq:Newton-Lyapunov}. The GADI scheme is
\begin{align}
\begin{cases}
(\alpha_k I+ A_k^*) X_{k+1,\ell+\frac{1}{2}}=X_{k+1,\ell}(\alpha_{k} I-A_k)+F(X_k),\\
X_{k+1,\ell+1}(\alpha_{k} I+ A_k) =X_{k+1,\ell}[A_k-(1-\omega_k)\alpha_{k} I]
+(2-\omega_{k})\alpha_{k} X_{k+1,\ell+\frac{1}{2}},
\end{cases}
\label{eq:Newton-GADI}
\end{align}
where $\ell=0,1,\cdots$, $X_{k+1,0}=X_{k}$, $\alpha_{k} >0$ and $\omega_{k}\in [0,2)$.

Combining the Newton scheme with the GADI scheme, we have the Newton-GADI algorithm, summarized in Algorithm \ref{alg:Newton-GADI} below.

\begin{algorithm}[H]
\caption{(Newton-GADI Algorithm)}
\label{alg:Newton-GADI}
~~~Given the matrices $A,~K,~Q$, the integer numbers $k_{\max}>1$ and $\ell_{\max}>1$,
the outer and inter iteration tolerance $\varepsilon_{out}$
and $\varepsilon_{inn}$, respectively.

~~~Step 1. Compute $\beta=1+\|A\|_{\infty},~~B=A+\beta I$
and the initial matrix $X_0=X^{-1}$
by solving Lyapunov equation $B^*X+XB-2Q=0$.

~~~Step 2. Set $X_{k+1,0}=X_{k}$
and compute the matrices
\begin{align*}
A_k=KX_k-A,~~F(X_k)=X_k K X_k+Q.
\end{align*}

~~~Step 3. Solve the GADI iteration scheme \eqref{eq:Newton-GADI} to obtain $X_{k+1}=X_{k+1,\ell_k}$ such that
\begin{align*}
\|A^*X_{k+1,\ell_k}+X_{k+1,\ell_k}A-X_{k+1,\ell_k}KX_{k+1,\ell_k}+Q\|< \varepsilon_{inn}
~~\mbox{or}~~
\ell_k>\ell_{\max}.
\end{align*}

~~~Step 4. Compute the normalized residual
\begin{align*}
\mbox{NRes}(X_{k+1})
=\frac{\|A^*X_{k+1}+X_{k+1}A-X_{k+1}KX_{k+1}+Q\|_2}
{\|A^*X_{k+1}\|_2+\|X_{k+1}A\|_2+\|X_{k+1}KX_{k+1}\|_2+\|Q\|_2}.
\end{align*}

~~~Step 5. If $\mbox{NRes}(X_{k+1})< \epsilon_{out}$ or $k>k_{max}$,
then the approximate solution of the CARE \eqref{eq:CARE} is $\tilde{X}=X_{k+1}$;
else set $k=k+1$ and return to Step 2.
\end{algorithm}

\begin{remark}
From Algorithm \ref{alg:Newton-GADI}, it can be seen that when $\omega_{k}=0$, the GADI method naturally reduces to
the ADI method in \cite{E.1988iterative}. We adopt the method given in \cite{S.2011matrix} to select the  desired initial matrix $X_0$,
see Step 1 in Algorithm \ref{alg:Newton-GADI}.

\end{remark}

For large-scale Lyapunov equations at each Newton step,
it is important to control the accuracy of the solution to gain efficiency,
and wish to keep the overall fast convergence property of the Newton method.
The inexact Newton method proposed in \cite{F.T.E.2009inexact}
has shown a rigorous guideline for the termination of the inner iteration while the fast local rate of convergence is retained.
The Lyapunov equation can be approximately solved such that
\begin{align*}
&\|(A-KX_k)^*X_{k+1}+X_{k+1}(A-KX_k)+X_k K X_k+Q\|
\leq \eta_k\|\mathcal{R}(X_k)\|\\
&=\eta_k\|A^*X_k+X_kA-X_kKX_k+Q\|.
\end{align*}
Formally, the iteration scheme is determined by solving
\begin{align}
(A-KX_k)^*X_{k+1}+X_{k+1}(A-KX_k)+X_k K X_k+Q=R_k.
\label{eq:Inexact-Newton-CARE}
\end{align}
The following convergence property for the inexact Newton method is referenced in \cite{F.T.E.2009inexact}.

\begin{thm}
\cite{F.T.E.2009inexact}
For the CARE \eqref{eq:CARE}, assume that $(A,K)$ is stabilizable and $(A,Q)$ is detectable,
and $X_*\in\mathbb{C}^{n\times n}$ is the Hermitian positive semi-definite solution. If there exist $\tilde{\delta}>0$ and $\tilde{\eta}>0$
such that for an initial matrix $X_0\in\mathbb{C}^{n\times n}$ meeting
$\|X_0-X_*\|\leq \tilde{\delta}$, then the iteration sequence $X_k$ generated by \eqref{eq:Inexact-Newton-CARE}
converges to $X_*$ if the residual $R_k$ satisfies
\begin{align*}
\|R_k\|\leq \eta_k \|A^*X_k+X_kA-X_kKX_k+Q\|.
\end{align*}
The convergence rate is linear if $\eta_k\in (0,\tilde{\eta}]$.
It is super-linear if $\eta_k\rightarrow 0$ and is quadratic if
$$ \eta_k\leq K_{\eta} \|A^*X_k+X_kA-X_kKX_k+Q\|$$
for some $K_{\eta}>0$.
\end{thm}

As a consequence,
the inexact Newton-GADI method
is summarized in Algorithm \ref{alg:Inexact-Newton-GADI}. Obviously, this algorithm consists of approximating
the solution in the $k$-th Newton step for
a certain tolerance $\eta_k>0$. It decreases when $k$ increases,
\textit{e.g.}, set $\eta_k=1/(k^3+1)$ (see Section \ref{sec:NE} for details).

\begin{algorithm}[H]
\caption{ (Inexact Newton-GADI Algorithm)}
\label{alg:Inexact-Newton-GADI}
~~~Given the matrices $A,~K,~Q$, the integer numbers $k_{\max}>1$ and $\ell_{\max}>1$,
a sequence of positive numbers $\{\eta_k\}$ such that $\eta_{k+1}<\eta_k$ for all $k$,
the outer iteration tolerance $\varepsilon_{out}$.

~~~Step 1. Compute $\beta=1+\|A\|_{\infty},~~B=-(A+\beta I)$
and the initial matrix $X_0=X^{-1}$
by solving the Lyapunov equation $B^*X+XB-2Q=0$.

~~~Step 2. Set $X_{k+1,0}=X_{k}$
and compute the matrices
\begin{align*}
A_k=KX_k-A,~~F(X_k)=X_k K X_k+Q.
\end{align*}

~~~Step 3. Solve the GADI iteration scheme \eqref{eq:Newton-GADI} to obtain $X_{k+1}=X_{k+1,\ell_k}$
such that
\begin{align*}
\|R_k\|\leq \eta_k \|A^*X_k+X_kA-X_kKX_k+Q\|
~~\mbox{or}~~
\ell_k>\ell_{\max}.
\end{align*}

~~~Step 4. Compute the normalized residual
\begin{align*}
\mbox{NRes}(X_{k+1})
=\frac{\|A^*X_{k+1}+X_{k+1}A-X_{k+1}KX_{k+1}+Q\|_2}
{\|A^*X_{k+1}\|_2+\|X_{k+1}A\|_2+\|X_{k+1}KX_{k+1}\|_2+\|Q\|_2}.
\end{align*}

~~~Step 5. If $\mbox{NRes}(X_{k+1})< \epsilon_{out}$ or $k>k_{max}$,
then the approximate solution of the CARE \eqref{eq:CARE} is $\tilde{X}=X_{k+1}$;
else set $k=k+1$ and return to Step 2.
\end{algorithm}

\subsection{Convergence analysis}

In this subsection, we prove that the Newton-GADI scheme is convergent. The convergence analysis of the Inexact Newton-GADI scheme is similar and hence is omitted. We recall the definition and property of Kronecker product.

For $A_{1}=(a_{ij})\in\mathbb{C}^{n_A \times m_A}$
and $B_{1}\in\mathbb{C}^{n_B \times m_B}$,
the Kronecker product is defined as
\begin{align*}
A_{1}\otimes B_{1}=\begin{pmatrix}
a_{11}B_{1} & a_{12}B_{1} & \cdots & a_{1m_A}B_{1}\\
a_{21}B_{1} & a_{22}B_{1} & \cdots & a_{2m_A}B_{1}\\
\vdots &  \vdots  &         & \vdots\\
a_{n_A 1}B_{1} & a_{n_A2}B_{1} & \cdots & a_{n_Am_A}B_{1}
\end{pmatrix}
\in \mathbb{C}^{(n_An_B)\times (m_Am_B)}.
\end{align*}
The vectorization operator (abbreviated as vec): $\mathbb{C}^{m\times n}\rightarrow \mathbb{C}^{mn}$ is
\begin{align*}
\mbox{vec}(X_{1})=(x^{T}_1,x^{T}_2,\cdots,x^{T}_n)^{T}\in\mathbb{C}^{mn},
~~\mbox{with}~~X_{1}=(x_1,x_2,\cdots,x_n)\in\mathbb{C}^{m\times n}.
\end{align*}
A brief review of some properties related to Kronecker products is required.

\begin{proposition}
For $A_{1},~B_{1},~C,~D,~X_{1}$ with appropriate dimensions, we have

(i) $(A_{1}\otimes B_{1})(C\otimes D)=(A_{1}C)\otimes(B_{1}D)$;

(ii) $(A_{1}\otimes B_{1})^{-1}=A_{1}^{-1}\otimes B_{1}^{-1}$;

(iii) $\Lambda(A_{1}\otimes B_{1})=\{\lambda\mu:~\lambda\in\Lambda(A_{1}) ~\mbox{and}~
\mu\in \Lambda(B_{1})\}$;

(iv) $(A_{1}\otimes B_{1})x=\mbox{vec}(B_{1}^TX_{1}A_{1})$,~$\mbox{vec}(X_{1})=x$.
\end{proposition}
Thereby, we have the following convergence conclusion for the Newton-GADI method.

\begin{thm}
\label{thm:GADIconver}
For the CARE \eqref{eq:CARE}, assume that $(A,K)$ is stabilizable and $(A,Q)$ is detectable,
the parameters $\alpha_{k}>0$ and $0\leq \omega_{k}<2$.
Then, for any $k=0,1,\cdots$,
the iteration sequence $\{X_{k+1,\ell}\}^{\infty}_{\ell=0}$
defined by \eqref{eq:Newton-GADI} converges to $X_{k+1}$.
\end{thm}

\begin{proof}
Taking the operation ``vec" from the GADI iteration scheme \eqref{eq:Newton-GADI}
yields
\begin{align}
\begin{cases}
(\alpha_{k} I+ I\otimes A_k^*) \mbox{vec}(X_{k+1,\ell+\frac{1}{2}})
=(\alpha_{k} I-A^T_k\otimes I)\mbox{vec}(X_{k+1,\ell})+\mbox{vec}(F(X_k)),\\
(\alpha_{k} I+ A^T_k\otimes I)\mbox{vec}(X_{k+1,\ell+1})
=[A^T_k\otimes I-(1-\omega_{k})\alpha_{k} I]\mbox{vec}(X_{k+1,\ell})
+(2-\omega_{k})\alpha_{k} \cdot \mbox{vec}(X_{k+1,\ell+\frac{1}{2}}).
\end{cases}
\label{eq:CARE-GADI-vec-1}
\end{align}
Denote $x=\mbox{vec}(X)$ and
$f(X_k)=\mbox{vec}(F(X_k))$.
The iteration scheme \eqref{eq:CARE-GADI-vec-1} is equivalent to
\begin{align}
\begin{cases}
(\alpha_{k} I+ I\otimes A_k^*) x_{k+1,\ell+\frac{1}{2}}
=(\alpha_{k} I-A^T_k\otimes I) x_{k+1,\ell}+ f(X_k),\\
(\alpha_{k} I+ A^T_k\otimes I) x_{k+1,\ell+1}
=[A^T_k\otimes I-(1-\omega_{k})\alpha_{k} I] x_{k+1,\ell}
+(2-\omega_{k})\alpha_{k} x_{k+1,\ell+\frac{1}{2}}.
\end{cases}
\label{eq:CARE-GADI-vec}
\end{align}
As $\alpha_{k}>0$ and $A_k$ is stable,
then $\alpha_{k} I+ I\otimes A_k^*$ and $\alpha_{k} I+ A^T_k\otimes I$
are nonsingular. By \eqref{eq:CARE-GADI-vec}, we get
\begin{align*}
x_{k+1,\ell+1}
&=(\alpha_{k} I+ A^T_k\otimes I)^{-1}
\Big\{ \Big[A^T_k\otimes I-(1-\omega_{k})\alpha_{k} I \Big]
+(2-\omega_{k})\alpha_{k}
(\alpha_{k} I+ I\otimes A_k^*)^{-1}(\alpha_{k} I-A^T_k\otimes I)\Big\} \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~
x_{k+1,\ell}+ (2-\omega_{k})\alpha_{k} (\alpha_{k} I+ A^T_k\otimes I)^{-1}
(\alpha_{k} I+ I\otimes A_k^*)^{-1}f(X_k)\\
&=(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
\Big[\alpha_{k}^2 I+ (I\otimes A_k^*)( A^T_k\otimes I)
-(1-\omega_{k})\alpha_{k} C_k
\Big]
x_{k+1,\ell}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~
+ (2-\omega_{k})\alpha_{k} (\alpha_{k} I+ A^T_k\otimes I)^{-1}
(\alpha_{k} I+ I\otimes A_k^*)^{-1}f(X_k)
\end{align*}
where $C_k=I\otimes A_k^*+A^T_k\otimes I$.
Denote
\begin{align*}
&T_k(\alpha_{k},\omega_{k})=(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
\Big[\alpha_{k}^2 I+ (I\otimes A_k^*)( A^T_k\otimes I)
-(1-\omega_{k})\alpha_{k} C_k \Big],\\
&G_k(\alpha_{k},\omega_{k})=(2-\omega_{k})\alpha_{k} (\alpha_{k} I+ A^T_k\otimes I)^{-1}
(\alpha_{k} I+ I\otimes A_k^*)^{-1},
\end{align*}
thus
\begin{align}
x_{k+1,\ell+1}=T_k(\alpha_{k},\omega_{k})x_{k+1,\ell}+G_k(\alpha_{k},\omega_{k})f(X_k).
\label{eq:iteration scheme}
\end{align}
Next, we prove that the iteration matrix $\rho(T_k(\alpha_{k},\omega_{k}))<1$ for $0<\alpha_{k}$
and $0\leq \omega_{k}<2$. Note that
\begin{align*}
2\alpha_{k} C_k=
-(\alpha_{k} I- I\otimes A_k^*)(\alpha_{k} I- A^T_k\otimes I)
+(\alpha_{k} I+ I\otimes A_k^*)(\alpha_{k} I+ A^T_k\otimes I),
\end{align*}
then
\begin{align}
T_k(\alpha_{k},\omega_{k})&=(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
\Big[\alpha_{k}^2 I+ (I\otimes A_k^*)( A^T_k\otimes I)
-(1-\omega_{k})\alpha_{k} C_k \Big]\notag\\
&=(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
\Big[(\alpha_{k} I- I\otimes A_k^*)(\alpha_{k} I- A^T_k\otimes I)
+\omega_{k}\alpha_{k} C_k \Big]\notag\\
&=\frac{1}{2}\Big\{(2-\omega_{k})
(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
(\alpha_{k} I- I\otimes A_k^*)(\alpha_{k} I- A^T_k\otimes I)
+\omega_{k} I
\Big\}\notag\\
&=\frac{1}{2}[(2-\omega_{k})T_k(\alpha_{k})+\omega_{k} I]
\label{eq:T-alpha-omega}
\end{align}
with
\begin{align}
T_k(\alpha_{k})=(\alpha_{k} I+ A^T_k\otimes I)^{-1}(\alpha_{k} I+ I\otimes A_k^*)^{-1}
(\alpha_{k} I- I\otimes A_k^*)(\alpha_{k} I- A^T_k\otimes I).
\label{eq:T-alpha}
\end{align}
Using \eqref{eq:T-alpha-omega}, it is evident that
\begin{align*}
\lambda_i(T_k(\alpha_{k},\omega_{k}))
=\frac{1}{2}[(2-\omega_{k})\lambda_i(T_k(\alpha_{k}))+\omega_{k} ](i=1,2,\cdots,n),
\end{align*}
and
\begin{align*}
\rho(T_k(\alpha_{k},\omega_{k}))
\leq \frac{1}{2}[(2-\omega_{k})\rho(T_k(\alpha_{k}))+\omega_{k} ].
\end{align*}
Obviously, the matrix $T_k(\alpha_{k})$ defined in \eqref{eq:T-alpha} is similar to
\begin{align*}
\hat{T}_k(\alpha_{k})=(\alpha_{k} I+ I\otimes A_k^*)^{-1}
(\alpha_{k} I- I\otimes A_k^*)(\alpha_{k} I- A^T_k\otimes I)
(\alpha_{k} I+ A^T_k\otimes I)^{-1}
\end{align*}
through the matrix $\alpha_{k} I+ A^T_k\otimes I$.
Thus we can obtain
\begin{align}
\rho(T_k(\alpha_{k}))
&\leq
\|(\alpha_{k} I+ I\otimes A_k^*)^{-1}(\alpha_{k} I- I\otimes A_k^*)\|_2\cdot
\|(\alpha_{k} I- A^T_k\otimes I)(\alpha_{k} I+ A^T_k\otimes I)^{-1}\|_2 \notag\\
&\leq
\|(\alpha_{k} I+ I\otimes A_k^*)^{-1}(\alpha_{k} I- I\otimes A_k^*)\|_2\cdot
\|(\alpha_{k} I- A^T_k\otimes I)(\alpha_{k} I+ A^T_k\otimes I)^{-1}\|_2 \notag\\
&=\|A^L_k\|_2\|A^R_k\|_2,
\label{eq:boundrhoALR}
\end{align}
where $A^L_k=(\alpha_{k} I+ I\otimes A_k^*)^{-1}(\alpha_{k} I- I\otimes A_k^*)$
and $A^R_k=(\alpha_{k} I- A^T_k\otimes I)(\alpha_{k} I+ A^T_k\otimes I)^{-1}$.
Then, for $\bm{x}\in\mathbb{C}^{n^2\times 1}$, we get
\begin{align*}
\|A^L_k\|^2_2
&=\max_{\|\bm{x}\|_2=1}
\frac{\|(\alpha_{k} I- I\otimes A_k^*)\bm{x}\|^2_2}
{\|(\alpha_{k} I+I\otimes A_k^*)\bm{x}\|^2_2}\\
&=\max_{\|\bm{x}\|_2=1}
\frac{\|(I\otimes A_k^*)\bm{x}\|^2_2-\alpha_{k} \bm{x}^*(I\otimes A_k+I\otimes A_k^*)\bm{x}+\alpha^2_k}
{\|(I\otimes A_k^*)\bm{x}\|^2_2+\alpha_{k} \bm{x}^*(I\otimes A_k+I\otimes A_k^*)\bm{x}+\alpha^2_k}\\
&\leq \max_{\|\bm{x}\|_2=1}
\frac{\|(I\otimes A_k^*)\bm{x}\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(I\otimes A_k))+\alpha^2_k}
{\|(I\otimes A_k^*)\bm{x}\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(I\otimes A_k))+\alpha^2_k}\\
&\leq \frac{\|I\otimes A_k^*\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(I\otimes A_k))+\alpha^2_k}
{\|I\otimes A_k^*\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(I\otimes A_k))+\alpha^2_k}\\
&= \frac{\|I\otimes A_k^*\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}
{\|I\otimes A_k^*\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}.
\end{align*}
Analogously, we obtain
\begin{align*}
\|A^R_k\|^2_2
\leq \frac{\|I\otimes A_k^T\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}
{\|I\otimes A_k^T\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}.
\end{align*}
Since $A_k$ is stable and $\alpha_{k}>0$, by \eqref{eq:boundrhoALR}, it follows that $\|A^L_k\|_2<1$ and $\|A^R_k\|_2<1$.
Therefore,
\begin{align*}
\rho(T_k(\alpha_{k}))< 1,~~ \mbox{and}~~
\rho(T_k(\alpha_{k},\omega_{k}))
\leq \frac{1}{2}[(2-\omega_{k})\rho(T_k(\alpha_{k}))+\omega_{k} ]<1,
\end{align*}
which implies that the iteration scheme \eqref{eq:iteration scheme} is convergent. The proof is completed.

\end{proof}

\subsection{Parameter selection}
\label{sec:parameter}

In this subsection, we first compare the convergence rate between the Newton-GADI and Newton-ADI methods.
That is, the relation between $\rho(T_k(\alpha_{k},\omega_{k}))$ and $\rho(T_k(\alpha_{k}))$ defined in Section 2.2 is discussed. Then we present a practical method  to select the quasi-optimal parameter.

\begin{thm}
\label{thm:relationADIandGADI}
Assume that the eigenvalues of $A_k$ have positive real parts.
For $j_1,~j_2\in\{1,\cdots,n^2\}$, denote
$$\rho(T_k(\alpha_{k},\omega_{k}))=\vert\zeta_{j_1}\vert=\vert a_{j_1}+i\, b_{j_1} \vert,~~\rho(T_k(\alpha_{k}))=\vert \eta_{j_2}\vert=\vert c_{j_2}+i\, d_{j_2} \vert.$$

(i) When $\vert \eta_{j_2}\vert^2\leq  c_{j_2}$, we have

\begin{align*}
\rho(T_k(\alpha_{k}))<\rho(T_k(\alpha_{k},\omega_{k}))<1.
\end{align*}

(ii) When $\vert \eta_{j_1}\vert^2> c_{j_1}$ and $0< \omega_{k} <\frac{4(\vert \eta_{j_1} \vert^2-c_{j_1})}{(1-c_{j_1})^2+d_{j_1}^2}<2,$
we have
\begin{align*}
\rho(T_k(\alpha_{k},\omega_{k}))<\rho(T_k(\alpha_{k}))<1.
\end{align*}
\end{thm}

\begin{proof}
For $\zeta_j=a_j+ib_j\in\Lambda(T_k(\alpha_{k},\omega_{k}))$, $\eta_j=c_j+id_j\in\Lambda(T_k(\alpha_{k}))(j=1,2,\cdots,n)$, then
\begin{align*}
2\vert\zeta_j \vert=\vert (2-\omega_{k})\eta_j +\omega_{k}\vert.
\end{align*}
Therefore,
\begin{align*}
4\vert\zeta_j \vert^2=\vert (2-\omega_{k})\eta_j +\omega_{k}\vert^2
=[(2-\omega_{k})c_j+\omega_{k}]^2+[(2-\omega_{k})d_j]^2
=(2-\omega_{k})^2\vert \eta_j\vert^2+\omega_{k}^2+2(2-\omega_{k})\omega_{k} c_j,
\end{align*}
\textit{i.e.,}
\begin{align*}
4\vert\zeta_j \vert^2-4\vert\eta_j \vert^2
&=(\omega_{k}^2-4\omega_{k})\vert \eta_j \vert^2+\omega_{k}^2+2(2-\omega_{k})\omega_{k} c_j\\
&=(1-2c_j+\vert\eta \vert^2)\omega_{k}^2-4(\vert \eta_j \vert^2-c_j)\omega_{k}\\
&=[(1-c_j)^2+d_j^2]\omega_{k}^2-4(\vert \eta_j \vert^2-c_j)\omega_{k}.
\end{align*}
Note that $$\rho(T_k(\alpha_{k},\omega_{k}))=\vert\zeta_{j_1}\vert,~~\rho(T_k(\alpha_{k}))=\vert \eta_{j_2}\vert.$$

(i) When $\vert \eta_{j_2}\vert^2\leq  c_{j_2}$, we have
\begin{align*}
\rho(T_k(\alpha_{k}))<\rho(T_k(\alpha_{k},\omega_{k}))<1.
\end{align*}

(ii) When $\vert \eta_{j_1}\vert^2> c_{j_1}$ and $0< \omega_{k} <\frac{4(\vert \eta_{j_1} \vert^2-c_{j_1})}{(1-c_{j_1})^2+d_{j_1}^2}<2,$ we have
\begin{align*}
\vert\zeta_{j_1} \vert<\vert \eta_{j_1}\vert,
\end{align*}
which means that $\rho(T_k(\alpha_{k},\omega_{k}))<\rho(T_k(\alpha_{k}))<1$.
\end{proof}

\begin{remark}
By Theorem \ref{thm:relationADIandGADI}, if $\vert \eta_{j_1}\vert^2> c_{j_1}$ and $0< \omega_{k} <\frac{4(\vert \eta_{j_1} \vert^2-c_{j_1})}{(1-c_{j_1})^2+d_{j_1}^2}<2,$ the Newton-GADI has a faster convergence rate than Newton-ADI proposed in \cite{E.1988iterative}. This is verified by numerical examples in Section \ref{sec:NE}.
\end{remark}

Next, we provide a method for selecting theoretical quasi-parameters by minimizing $\rho(T_k(\alpha_{k},\omega_{k}))$.
\begin{thm}
\label{thm:para}
Assume that the eigenvalues of $A_k$ have positive real parts, set
\begin{align}
\nu_n=\max_{j}\sigma_j(A_k), ~~
\min_j Re(\lambda_j(A_k))=Re(\mu_1)=Re(a_\mu+i b_\mu)=a_\mu.
\label{notation:numu}
\end{align}
Then the quasi-optimal parameter $\alpha_k$ can be obtained by
\begin{align}
\alpha_{k}^*=\mbox{arg}\min_{\alpha_{k}}
\frac{\nu_n^2- 2\alpha_{k} a_\mu+\alpha^2_k}
{\nu_n^2+2\alpha_{k} a_\mu+\alpha^2_k}
=\nu_n.
\label{para:alpha}
\end{align}
\end{thm}

\begin{proof}
Using the bound defined by \eqref{eq:boundrhoALR}, we have
\begin{align*}
\rho^2(T_k(\alpha_{k}))
&\leq \|A_L\|^2_2\|A_R\|^2_2\\
&\leq \frac{\|I\otimes A_k^*\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}
{\|I\otimes A_k^*\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k} \cdot
\frac{\|I\otimes A_k^T\|^2_2- 2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}
{\|I\otimes A_k^T\|^2_2+2\alpha_{k} \min_{i} Re(\lambda_i(A_k))+\alpha^2_k}\\
&=\frac{\nu_n^2- 2\alpha_{k} a_\mu+\alpha^2_k}
{\nu_n^2+2\alpha_{k} a_\mu+\alpha^2_k} \cdot
\frac{\nu_n^2- 2\alpha_{k}a_\mu+\alpha^2_k}
{\nu_n^2+2\alpha_{k}a_\mu +\alpha^2_k},
\end{align*}
\textit{i.e.,}
\begin{align*}
\rho(T_k(\alpha_{k}))\leq
\frac{\nu_n^2- 2\alpha_{k} a_\mu+\alpha^2_k}
{\nu_n^2+2\alpha_{k} a_\mu+\alpha^2_k},
\end{align*}
where $\nu_n$ and $a_\mu$ are given in \eqref{notation:numu}.
Denote
\begin{align*}
\Psi(\alpha_{k})=\frac{\nu_n^2- 2\alpha_{k} a_\mu+\alpha_{k}^2}
{\nu_n^2+2\alpha_{k} a_\mu+\alpha_{k}^2}.
\end{align*}
Then
\begin{align*}
\frac{\partial\Psi(\alpha_{k})}{\partial\alpha_{k}}
=\frac{4 a_\mu (\alpha_{k}^2-\nu_n^2)}
{(\nu_n^2+2\alpha_{k} a_\mu+\alpha_{k}^2)^2}.
\end{align*}
It is easy to show that $\Psi(\alpha_{k})$ gets to its minimum
when $\alpha_{k}=\nu_n$, \textit{i.e.,}
$$\alpha_{k}^*=\mbox{arg}\min\limits_{\alpha_k}\Psi(\alpha_k)=\nu_n,$$
where $\nu_n$ depends on the matrix $A_k$.
\end{proof}

\begin{remark}
In terms of Theorem \ref{thm:para}, we usually select $\omega_k=1$ and $\alpha_k^*$ given in \eqref{para:alpha}
as the quasi-optimal parameters. It can be seen that this parameter choice is reasonable using the residual analysis in Section \ref{sec:NE}.
\end{remark}

\section{Numerical experiments}
\label{sec:NE}

In this section, we use several examples of the quadratic optimal control
to show the numerical feasibility and effectiveness of the Newton-GADI and inexact Newton-GADI (InNewton-GADI) algorithms.
The whole process is performed on a computer with Intel Core 3.20GHz CPU, 4.00GB RAM and MATLAB R2017a. IT(inn) and IT(out) represent the number of inter iteration steps and
outer iteration steps, respectively. Denote IT(cumul) by the sum of inter and outer iteration steps,
IT(ave) by the number of the GADI iterative steps required by the average Newton step, and CPU by the computing time.
The notation $\mbox{Tri}(c,a,b)_{n\times n}$ demonstrates
the $n\times n$ size tridiagonal matrix
\begin{align*}
\begin{pmatrix}
a & b &        &  &\\
c & a & b      &   &\\
  & c & \ddots & \ddots&\\
  &   &\ddots  & \ddots &b\\
  &   &        &   c    &a
\end{pmatrix}.
\end{align*}

In our numerical experiment, set the outer iteration tolerance
$\varepsilon_{out}=1\times 10^{-8},~\varepsilon_{inn}=1\times 10^{-8}$,
$\ell_{\max}=1000,~k_{\max}=1000$.
In the inexact Newton-GADI algorithm, set $\eta_k=1/(k^3+1)$.
Denote the numerical Hermitian semi-definite solution as $\tilde{X}$, the finally normalized residual as
\begin{align*}
\mbox{NRes}(\tilde{X})
=\frac{\|A^*\tilde{X}+\tilde{X}A-\tilde{X}K\tilde{X}+Q\|_2}
{\|A^*\tilde{X}\|_2+\|\tilde{X}A\|_2+\|\tilde{X}K\tilde{X}\|_2+\|Q\|_2},
\end{align*}
and the error of the inter iteration in $k$-th Newton iteration step as
\begin{align*}
R(X_k)=A_k^*\tilde{X}^{in}_k+\tilde{X}^{in}_kA_k-X_kKX_k-Q,
\end{align*}
with $\tilde{X}^{in}_k$ is the numerical solution of Lyapunov equation
in $k$-th Newton iteration step.

When $\omega_{k}=0$, the Newton-GADI reduces to Newton-ADI proposed in \cite{E.1988iterative}.
It can be seen that the Newton-GADI converges faster than Newton-ADI in the following examples.

\begin{example}
Consider the linear time-invariant system of the form
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)+B\bm{u}(t)=\begin{pmatrix}
-2+10\,i & 0  & -1\\
0  & -1+10\,i & 0\\
-1 & -1 & -2\,i
\end{pmatrix}\bm{p}(t)
+\begin{pmatrix}
-2 & 0 & -1\\
0 & -1 & -1\\
1 & 0 & -2
\end{pmatrix}\bm{u}(t).
\end{align*}
We wish to minimize the cost function $J$ defined in \eqref{eq:def-cost-J}.
The cost penalty matrices are
\begin{align*}
R = \begin{pmatrix}
1 & 0  & 0\\
0  & 1 & 0\\
0 & 0 & 4
\end{pmatrix},~
Q=\begin{pmatrix}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 5
\end{pmatrix}.
\end{align*}
Then
$$
K=BR^{-1}B^{T}=\begin{pmatrix}
17/4 & 1/4 & -3/2\\
1/4  & 5/4 & 1/2\\
-3/2 & 1/2 & 2
\end{pmatrix}.
$$
\end{example}
We can verify that $K$ is positive definite and $Q$ is semi-positive definite.
The pair $(A,K)$ is stabilize as
\begin{align*}
\Lambda(A-KI_3)=\{-1.9936-1.9179\,i, -6.2704+9.9858\,i, -2.2360 + 9.9321\,i\}.
\end{align*}
The pair $(A,Q)$ is detectable as
\begin{align*}
\Lambda(Q^*-A^*F)=\{  -2.0000 + 0.0000\,i, -3.5000 - 1.6583\,i, -3.5000 + 1.6583\,i\}
\end{align*}
with
\begin{align*}
F=\begin{pmatrix}
-0.2785 - 0.5729\,i &  0 &  0.2387 - 0.3660\,i\\
-0.1651 - 0.0121\,i & -0.0297 - 0.2970\,i &  0.2773 - 0.3460\,i\\
0.2865 - 1.6393\,i &  0 &  0.1830 + 3.1194\,i
\end{pmatrix}.
\end{align*}
The numerical Hermitian semi-definite solution is
\begin{align*}
\tilde{X}= \begin{pmatrix}
-0.0020+ 0.0120i & -0.0034 + 0.0121i  &-0.1269-0.0611i\\
-0.0034 + 0.0121i&   0.0001 + 0.0618i &-0.1251-0.0780i\\
-0.1269 - 0.0611i&  -0.1251 - 0.0780i & 1.1794-0.9689i
\end{pmatrix}.
\end{align*}

We know that the approximate solution of the Lyapunov equation
can be obtained by using the inexact algorithm at each Newton step.
When the error generated by the inexact algorithm is very large, it will affect
the convergence rate in next step.
Figure \ref{fig1:errorinn} displays the error $\|R(X_k)\|_2$ of
the inexact inter iterative algorithm in each Newton step.
Moreover, due to the large error $\|R(X_2)\|_2$ caused by the GADI in the second iteration step,
from Figure \ref{fig1:IT}, we know that the inner iteration step exceeds the maximum and is not convergent in the third Newton step.
Therefore, we reset $\eta_k=\frac{1}{k^4+1}$ in the inexact Newton-ADI algorithm,
see Figure \ref{fig1:errorinn2}.

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG1_compare_innR.png}}
\caption{Comparing the inner iteration accuracy of two inexact algorithm with $\eta_k=\frac{1}{k^3+1}$.}\label{fig1:errorinn}
\end{figure}

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG1_compare_IT.png}}
\caption{Comparing the number of inner iteration steps of
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI at each Newton step
with $\eta_k=\frac{1}{k^3+1}$.}\label{fig1:IT}
\end{figure}

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG1_compare_innR2.png}}
\caption{Comparing two inexact algorithms.}\label{fig1:errorinn2}
\end{figure}

Figure \ref{fig1:error2} shows the convergence process of the four algorithms,
including the Newton-ADI ($\omega=0$), the Newton-GADI,  the InNewton-ADI ($\omega=0$)
and the InNewton-GADI methods.
It is not difficult to find that these algorithms
have almost the same convergence rate.
\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG1_compare_NRes2.png}}
\caption{Comparing the convergence rate of the outer iteration for
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI.}\label{fig1:error2}
\end{figure}

Figure \ref{fig1:IT2} shows that the number of the GADI steps
required for each Newton step. It is easy to see that the inexact Newton-GADI method
has the least number of inter iteration steps among the four algorithms. The total number of steps in the convergence process
is the least compared other methods, see Table \ref{tab:EG1}
for details.

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG1_compare_IT2.png}}
\caption{Comparing the number of inner iteration steps of
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI at each Newton step.}\label{fig1:IT2}
\end{figure}

\begin{table}[!hptb]
\vspace{-0.4cm}
\centering
\caption{Numerical results of Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI.}\label{tab:EG1}
\vspace{0.2cm}
\renewcommand\arraystretch{1.3}
\setlength{\tabcolsep}{3mm}
{\begin{tabular}{c|c|c|c|c|c}\hline
             &$\omega_k^*$  &  $NRes(\tilde{X})$ &IT(out)    & IT(ave)  &IT(cumul)   \\ \hline
Newton-ADI   & 0          &8.1384e-09          &19          &48.21     & 916\\ \hline
Newton-GADI  & 1       & 8.2030e-09            & 19         &30.16     & 573    \\ \hline
InNewton-ADI & 0          & 8.4516e-09         & 26       &26.32       & 500  \\ \hline
InNewton-GADI &1          & 6.6070e-09         & 20       &14.30       & 286  \\ \hline
\end{tabular}}
\end{table}

\begin{example}
Consider the  linear time-invariant system of the form
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)+B\bm{u}(t)
=\begin{pmatrix}
0 & -1  & 0 & 0\\
1  &0  & -1 & 0\\
0 & 1  &  0  &-1\\
0 & 0  &  1  & 0
\end{pmatrix}\bm{p}(t)
+10^{-3}\times \begin{pmatrix}
    3  & -50 &   1 &   2\\
   1  & -3 &  -2&   1\\
   -3 &  1&  3 &   4\\
   3 & -1&  -4 &  3
\end{pmatrix}\bm{u}(t).
\end{align*}
We wish to minimize the cost function $J$ defined in \eqref{eq:def-cost-J}.
The cost penalty matrices are
\begin{align*}
R = I_4,~
Q= \begin{pmatrix}
    0.0025  &       0 &        0  &       0\\
         0  &  0.0111 &   0.0025  &       0\\
         0  &  0.0025 &   1.0006  &  0.0200\\
         0  &       0 &   0.0200  &  0.0004
\end{pmatrix},~~\mbox{therefore}~~K=BB^{T}.
\end{align*}
\end{example}

It is easy see that $K$ is semi-positive definite and
$Q$ is semi-positive definite. The pair $(A,K)$ is stabilize as
\begin{align*}
\Lambda(A-KI_4)=\{-0.0004 + 1.6180\,i, -0.0004 - 1.6180\,i, -0.0009 + 0.6180\,i,-0.0009 - 0.6180\,i\}.
\end{align*}
The pair $(A,Q)$ is detectable as
\begin{align*}
\Lambda(Q^*-A^*F)=\{
  -0.0013 +50.0608\,i,
  -0.0013 -50.0608\,i,
  -0.0009 + 7.6862\,i,
  -0.0009 - 7.6862i\,\}
\end{align*}
with
\begin{align*}
F=\begin{pmatrix}
   17.5243 &   0.3931 &  -8.2505 &  -0.3925\\
    0.3931 &  25.8550 &   0.3901 &  -8.2707\\
   -8.2505 &   0.3901 &  25.8338 &  -0.0035\\
   -0.3925 &  -8.2707 &  -0.0035 &  17.5505
\end{pmatrix}.
\end{align*}
The numerical Hermitian semi-definite solution is
\begin{align*}
\tilde{X}= \begin{pmatrix}
   17.4818 &   0.3916 &  -8.2438 &  -0.3924\\
    0.3916 &  25.8038 &   0.3901 &  -8.2638\\
   -8.2438 &   0.3901 &  25.7818 &  -0.0035\\
   -0.3924 &  -8.2638 &  -0.0035 &  17.5055
\end{pmatrix}.
\end{align*}

Figure \ref{fign4:errorinn} displays that the error of
the inexact inter iterative algorithm with the increase of Newton step.
Figure \ref{fign4:error} shows the convergence process of the four algorithms and
reports that these algorithms have almost the same convergence rate.
\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EGn4_compare_innR.png}}
\caption{Comparing the inner iteration accuracy of two inexact algorithm with $\eta_k=\frac{1}{k^4+1}$.}\label{fign4:errorinn}
\end{figure}

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EGn4_compare_NRes.png}}
\caption{Comparing the convergence rate of the outer iteration for
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI with $\eta_k=\frac{1}{k^4+1}$ .}\label{fign4:error}
\end{figure}

Figure \ref{fign4:IT} reports the number of the GADI steps
required for each Newton step. It can be seen that the InNewton-GADI algorithm has a clear advantage over other algorithms.

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EGn4_compare_IT.png}}
\caption{Comparing the number of inner iteration steps of
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI at each Newton step.}\label{fign4:IT}
\end{figure}

Table \ref{tabn4:EG} displays the numerical results of these four algorithms.
It is not difficult to find that this example is ill-conditioned. Some existing ADI algorithms may be convergent slowly or not convergent, while the GADI scheme proposed in this paper
can accelerate the convergence process.
\begin{table}[!hptb]
\vspace{-0.4cm}
\centering
\caption{Numerical results of Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI.}\label{tabn4:EG}
\vspace{0.2cm}
\renewcommand\arraystretch{1.3}
\setlength{\tabcolsep}{3mm}
{\begin{tabular}{c|c|c|c|c|c}\hline
             &$\omega_k^*$  &  $NRes(\tilde{X})$ &IT(out)    & IT(ave)  &IT(cumul)   \\ \hline
Newton-ADI   & 0          &9.5234e-09          &15          &204.40     &3066\\ \hline
Newton-GADI  & 0.015       &9.5846e-09          &15         &193.80    & 2907    \\ \hline
InNewton-ADI & 0          & 1.2046e-10         & 15       &175.87       & 2638  \\ \hline
InNewton-GADI &0.015       &3.5208e-09         & 15       &153.00       &2295  \\ \hline
\end{tabular}}
\end{table}

\begin{example}
Consider the linear time-invariant system of the form
\begin{align*}
\dot{\bm{p}}(t)=A\bm{p}(t)+B\bm{u}(t)
\end{align*}
where
\begin{align*}
A=\mbox{Tri}(-1-r,-4+8\,i,-1+r)_{n\times n},~
b=(1,0,\cdots,0)\in\mathbb{R}^{1\times n},~~B=(b^T, I_n),
\end{align*}
with $r=\dfrac{1}{2n+2}$.
To go further, we need to minimize the cost function $J$ defined in \eqref{eq:def-cost-J}.
The cost penalty matrices are $R=I_n$,~$K=BR^{-1}B^{T}$,~$Q=c^{T}c$ with $c=(\dfrac{1}{\sqrt{10}},0,\cdots,0)\in\mathbb{R}^{1\times n}$.
\end{example}

When $n=128$, Figure \ref{fig2:errorinn} displays that the error of
the inexact inter iterative algorithm with the increase of Newton step.
\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG2_compare_innR128.png}}
\caption{Comparing the inner iteration accuracy of two inexact algorithm with $\eta_k=\frac{1}{k^3+1}$ and $n=128$.}\label{fig2:errorinn}
\end{figure}

Figure \ref{fig2:error} shows the convergence process of the four algorithms and
reports that these algorithms have almost the same convergence rate.
Moreover, it is easy to find that
the convergence rate of the iteration is slow in the beginning.
From the fifth iteration step, these algorithms begin to converge rapidly.
\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG2_compare_NRes128.png}}
\caption{When $n=128$, comparing the convergence rate of the outer iteration for
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI with $\eta_k=\frac{1}{k^3+1}$ .}\label{fig2:error}
\end{figure}

Figure \ref{fig2:IT} reports the number of the GADI steps
required for each Newton step. It can be seen that the InNewton-GADI algorithm has a clear advantage over other algorithms.

\begin{figure}[!htbp]
\centering
\subfigure{
\includegraphics[width=6in]{EG2_compare_IT128.png}}
\caption{When $n=128$, comparing the number of inner iteration steps of
Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI at each Newton step.}\label{fig2:IT}
\end{figure}

Table \ref{tab:EG2} displays the numerical results of these four algorithms
when $n=64,~n=128,~n=512$ and $n=1024$.
We can see that as $n$ increases, the number of iterative steps and the finally normalized residual $NRes(\tilde{X})$ of each algorithm are almost the same,
while the CPU time increases gradually.
\begin{table}[!hptb]
\vspace{-0.4cm}
	\centering
	\caption{Numerical results of Newton-ADI, Newton-GADI, InNewton-ADI and InNewton-GADI
as $n$ increases.}\label{tab:EG2}
    \vspace{0.2cm}
	\renewcommand\arraystretch{1.3}
	\setlength{\tabcolsep}{3mm}
	{\begin{tabular}{c|c|c|c|c|c|c|c}\hline
$n$               &  Algorithm  &$\omega_k^*$  & $NRes(\tilde{X})$                &IT(out)    & IT(ave)  &IT(cumul) & CPU(s)  \\ \hline
\multirow{4}*{64} &Newton-ADI & 0          &6.3978e-09          &11    &12.73     & 140  & 0.2140 \\ \cline{2-8}
                &Newton-GADI  & 1       & 9.4256e-09            & 7     &14.71     &103  & 0.1320 \\ \cline{2-8}
                &InNewton-ADI & 0          &1.1340e-09         &8       &8.13       &65  & 0.1070\\ \cline{2-8}
                 &InNewton-GADI &1          &4.2259e-09        &8       &4.50       &36  & 0.0830\\ \hline
\multirow{4}*{128}  &Newton-ADI   & 0     &8.4557e-09          &9        &15.44     &139 &0.7540  \\ \cline{2-8}
                    &Newton-GADI  & 1       & 9.4644e-09        & 7    &14.72       &103 &0.5400   \\ \cline{2-8}
                   &InNewton-ADI & 0          &1.1531e-09         & 8    &8.13      &65  &0.5150  \\ \cline{2-8}
                   &InNewton-GADI &1          &4.2424e-09         & 8     &4.50     &36  &0.3680  \\ \hline
 \multirow{4}*{512} &Newton-ADI   & 0     &8.4857e-09          &9      &15.44     &139  &22.1390 \\ \cline{2-8}
                    &Newton-GADI  & 1       & 9.4771e-09       & 7   &14.72     & 103  &16.9060    \\ \cline{2-8}
                   &InNewton-ADI & 0          &1.1596e-09      & 8     &8.13    &65    &15.5520 \\ \cline{2-8}
                  &InNewton-GADI &1      & 4.2480e-09         & 8      &4.50    & 36   &10.8330 \\ \hline
\multirow{4}*{1024} &Newton-ADI   & 0     &8.4872e-09          &9      &15.44     &139  &256.4320 \\ \cline{2-8}
                    &Newton-GADI  & 1       & 9.4777e-09       & 7   &14.72     & 103  &201.9130    \\ \cline{2-8}
                   &InNewton-ADI & 0          &1.1600e-09      & 8     &8.13    &65  &159.0060 \\ \cline{2-8}
                  &InNewton-GADI &1      & 4.2484e-09         & 8      &4.50    & 36 &113.2540 \\ \hline
\end{tabular}}
\end{table}

\section{Conclusion}

In this paper, we have proposed the Newton-GADI algorithm to solve the complex CARE. Moreover, by inaccurately solving a Lyapunov equation in each Newton step,
 the inexact Newton-GADI algorithm is presented.
Finally, we have offered numerical experiments to compare the effectiveness.
It can be seen that the Newton-GADI method has a faster inter iterative convergence rate
than the Newton-ADI method, while the inexact Newton-GADI method is more efficient.
Therefore, the Newton-GADI and inexact Newton-GADI methods could be valid and attractive algorithms
to solve the ill-conditioned Riccati equation.
In the future, we will further study parameter selection and solving the low-rank large-scale CARE.

\nocite{*

redpajama_set_name: RedPajamaArXiv


Gujarat board Secondary School Certificate (SSC) had information regarding the class 10th examination results 2015 will be declared to announced on June 4 at 6 AM.
Students desiring to check their results will have to log on to the given website and provide their roll number to access the results.
Exams for class 10 of Gujarat Board for the year 2015, were conducted in the month of March.
Every year, more than 9 lakh students appear for the exams which are a stepping stone for students in creating a career for themselves.
GSEB was formed on May 1, 1972. Gujarat State Board is responsible for preparing syllabus, recommending books, and conducting the exams for all school levels in Gujarat among other things.
From below sites to download the gujarat SSC 10th class examination result 2015.


text:
NBA insider shares surprising information about Dwyane Wade
Dwyane Wade might not get what he wanted from Cleveland. Image Credit: Erik Drost / Flickr
NBA Insider David Aldridge has some interesting news regarding Dwyane Wade.
by Asmir Pekmic (article) and Milos Kitanovic (video)
October 2, 2017 at 12:50 PM Updated on October 5, 2017 at 9:39 AM
Dwyane Wade reunited with LeBron James last week as he accepted a one-year contract with the Cleveland Cavaliers. During their time together in Miami, Wade and James were one of the most entertaining duos to watch in the league. Not only were they entertaining, but they were also very successful as they won two championships and made the NBA Finals four straight times.
After he reached a buyout agreement with the Chicago Bulls, Wade had plenty of teams to choose from. Beside the Cavaliers, the Miami Heat and the San Antonio Spurs were interested in his services. The shooting guard was also recruited by Carmelo Anthony in an attempt to get him to sign with the Oklahoma City Thunder.
The biggest reason why Wade chose the Cavaliers over the other teams was LeBron James. Their friendship played a big role in Wade's decision, but there is another reason as well. Apparently, the Cavaliers told Wade he would start if he signs with them. However, that may not be the case.
Dwyane Wade's role in Cleveland
Dwyane Wade is a three-time NBA champion and one of the best shooting guards to ever play in the league. He was brought to Cleveland for a reason and the Cavaliers will definitely benefit from his presence. The veteran guard is expected to help Cleveland beat the Golden State Warriors and win another championship.
However, he might be forced to help the team off the bench. David Aldridge of TNT tweeted that Dwyane Wade will most likely be a backup shooting guard and J.R.
Smith will be a starter.
Do not know for sure but would think this means @DwyaneWade will come off bench as well and @TheRealJRSmith starts at the two.
— David Aldridge (@daldridgetnt) October 2, 2017
With Wade on the bench, the Cavaliers would have Derrick Rose, Smith, LeBron James, Jae Crowder and Kevin Love in the starting lineup. This lineup would provide maximum spacing and the Cavaliers would most likely use it until Isaiah Thomas comes back.
On the other side, Dwyane Wade would lead the second unit. Beside him, Tristan Thompson would be a backup center while Iman Shumpert and Kyle Korver would be backup wings. Cleveland's depth chart is not yet clear, but there's a good chance that Jeff Green will be in the second unit as well.
Kevin Love or Tristan Thompson
J.R. Smith and Dwyane Wade are not the only players on the Cavaliers who are fighting for a starting spot.
Tristan Thompson, who was the Cavaliers' starting center for the past few years, will most likely be replaced by Kevin Love.
Moving Thompson to the bench would allow Cleveland to use both LeBron James and Jae Crowder in the starting lineup, which would give them more spacing and athleticism. Considering that Love is a great rebounder, he will probably play well as a center.
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I love watching basketball and writing about it, especially about my favorite team, Miami Heat. I'm an avid NBA fan and I love to express my opinion about the league and its teams.
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text:
Help Decide the Winner of OnPoint's $5,000 Community Builder Award
Community Impact Story Article « Back to blog
Since the pandemic began, we have seen outstanding teachers and students across our community taking learning to new heights to keep each other engaged, healthy and energized. Each year, the OnPoint Prize for Excellence in Education celebrates local educators who are shattering the mold for traditional learning and enriching the lives of their students.
In addition to awarding outstanding educators, the OnPoint Prize for Excellence in Education also provides awards for impactful school projects. As students return to the classroom for hybrid learning, inspiring creativity and fostering community will be more important than ever. We are proud, once again, to support five new Community Builder projects for schools across Oregon and Southwest Washington.
The nomination period for the OnPoint Prize for Excellence in Education began on February 23 and closed on April 13. From all the Community Builder project nominations, we are proud to announce our nine finalists who will be voted on by you, our online community. The winning school, selected by community votes, will receive a $5,000 donation, while four other schools chosen by the OnPoint Prize Selection Committee will each receive $2,000.
The voting period runs now through Tuesday, May 4. We encourage you to read about the schools below and cast your vote for the project you think will make the most of the $5,000 award. Then, look for our announcement of the Community Builder Award winners and the six finalists for the OnPoint Prize Educator of the Year on May 11.
Learn more about this year's finalists:
Awbrey Park Elementary School's Conscious Discipline Program
Eugene School District, Eugene, OR
Awbrey Park Elementary School's Conscious Discipline Program is an evidence-based, trauma-informed approach to supporting teachers' and students' social and emotional needs in the classroom. The program provides various behavior management strategies and classroom structures that teachers can use to turn everyday situations into learning opportunities. Conscious Discipline meets CASEL's SELect Program designation—the highest designation for evidence-based programs—in the CASEL Guide to Effective Social and Emotional Learning Programs. This designation indicates that Conscious Discipline can play a central role in a school's approach to promoting students' social and emotional learning.
The Eugene School District is allotting funds for Conscious Discipline classroom materials and training for approximately 30 Title One kindergarten teachers in June 2021. With the additional Community Builder funds, the training at Awbrey Park would also include teachers and students in first and fifth grades.
Cheldelin Middle School's Fire and Food Courtyard
Corvallis School District, Corvallis, OR
Cheldelin Middle School believes that life's strongest connections form over food and fire. To create a unique learning experience for students that links scientific measurement, mathematical construction principles and art creation, Cheldelin Middle School seeks to build an outdoor space equipped with a wood-fired kiln. Built above the kiln's firebox will be a bread oven. Through the power of fire and food, Cheldelin's outdoor space will spark cross-curricular learning, create lasting positive memories, foster student pride and create a new recreational activity for the community.
Cheldelin Middle School would use the Community Builder funds to purchase a metal shed roof, the first item needed to begin the project.
David Douglas High School's Scots Readers
David Douglas School District, Portland, OR
David Douglas School District teachers have found reading skills in elementary school are a key indicator for high school success. Right now, reading levels at the district's elementary schools are well below average. David Douglas High School's Scots Readers program hopes to change that by pairing district second graders with high school athletes to share and read their favorite books.
The Scots Readers program's goal is to inspire second graders to recognize the importance of reading, create mentorship opportunities for high school students, and strengthen community across a district with over 60 languages spoken.
The Scots Readers program will culminate in a David Douglas High School basketball event in January 2022. Second graders will receive an "I am a Scots Reader" t-shirt, poster, three books, and a literacy handbook to recognize their achievement at the game. They will also have the opportunity to meet the boys and girls varsity basketball teams
David Douglas High School would use Community Builder funds to pay for the second graders' "I am a Scots Reader" t-shirts.
Holy Cross Catholic School's Outdoor Enhancement Project
Archdiocese of Portland, Portland, OR
Holy Cross Catholic School has been hard at work over the last three years on a three-phase outdoor enhancement project. Holy Cross Catholic School recently completed the first phase of its new and improved play area, which included installing multiple new play structures, a music area and benches.
The school began work on its Hodge Street Garden two and half years ago, introducing gardening, agriculture and science learning opportunities for students, teachers, parents, youth groups and the surrounding community. So far, the school community has worked together to create a pollinator garden, hummingbird sanctuary and worm composting. Through the pandemic, the Hodge Street Garden has allowed the school community to continue gathering safely as the enhanced outside space provides enough room for physical distancing.
The third and final phase of Holy Cross Catholic School's campus enhancement project is its Environmental Garden and West Gym area. This final phase will grade, plumb and install raised beds, pavers and benches. When complete, the Environmental Garden will be an area for classrooms to adopt garden beds, conduct outdoor learning, create outdoor space for teachers to eat lunch and establish a peaceful place for the neighbors to enjoy as well.
Holy Cross Catholic School would use the Community Builder funds to complete the Hodge Street Garden and begin work on the Environmental Garden and West Gym area.
Holy Family Catholic School's Celebration of International Diversity
Holy Family Catholic School seeks to supports faculty, students, families and the greater parish in exploring prejudices, racial diversity and solutions that can bring about a more equitable and just community.
Led by the Racial Equity Committee, Holy Family Catholic School's diversity equity and inclusion project will naturally foster a safe learning environment and sense of belonging. In celebration of the diversity of cultures, religions, languages and races both in and out of the school community, the school will offer guest speakers or authors from different cultures, a social justice book club for families, students and faculty, cooking classes and art projects. Holy Family Catholic School's social justice activities will culminate in its annual International Fair in May.
The school would use the Community Builder funds to engage speakers, provide books for the book club, purchase food for cooking classes and purchase supplies for art projects.
Lyle Elementary School's Recess for All
Dallas School District 2, Dallas, OR
The Recess for All Project seeks to redesign Lyle Elementary School's recess to better serve its students. Lyle Elementary School strives to go beyond the typical recess activities and make recess a time for all students to engage in activities that will benefit their physical and mental health and prepare them to return to the classroom ready to learn.
Recess for All will diversify recess activities by creating spaces for group and individual play, add new activities for different levels of play, and define spaces so students can easily identify which area they would like to be in during their recess time.
Recess for All will benefit not only Lyle students but also the community at large. Located near
downtown, the school is next to the Lyle Sports Complex and Dallas City Park, a popular area for many in the Dallas community. Because school grounds are open to the public outside of school hours, children of all ages will utilize the improved recess area.
Lyle would use the Community Builder funds to purchase new small group and individual activity stations, signage, barriers to better divide up the recess area, and a storage structure to help house recess materials.
Maple Elementary's Bulb Planting Project
Springfield School District, Springfield, OR
After spending most of the 2020-21 school year apart, Maple Elementary's bulb planting project will welcome students back in the fall of 2021 by bringing them together to work on a common goal benefitting the school and the entire neighborhood.
When school starts in September 2021, students and families will work together to plant bulbs in the school's front entry area. When spring 2022 arrives, the school and community will enjoy the beauty of the work they did in the fall. Because the bulbs will bloom annually, they will beautify the area for years to come.
Additionally, in the time of COVID, having outdoor learning spaces available is more critical than ever. Teachers and students would use these garden spaces for science units, inspiration for art projects and creative writing, and a calm space where students can take a break. After a year spent online, these hands-on learning opportunities will be integral to students' education.
Maple Elementary School would use the Community Builder funds to purchase enough bulbs for the school community and the gardening supplies necessary to allow the event to be open to all families. Bulbs varietals will include tulips, daffodils, hyacinths and camas, a flower highly valued by the local Kalapuya tribe.
St. Thomas Academy's Outdoor Grounds Enhancement
Diocese of Baker, Redmond, OR
Since opening for in-person learning in fall 2020, St. Thomas Academy has been immersed in finding the safest way to offer instruction and engage students through the pandemic. In addition to requiring masks, practicing physical distancing, creating class cohorts, encouraging handwashing, and installing health checkpoint stations, St. Thomas Academy also understands the safest place to be right now is outside.
To bring more multi-subject learning outdoors and allow students and teachers to reap the many health and learning benefits of being outside, the school would use the Community Builder funds to build a permanent high-quality gazebo and four picnic tables and benches. The gazebo would be located near the school's new playground, which is open during non-school hours for the community to use.
Vern Patrick Elementary School's StoryWalk®
Redmond School District 2J, Redmond, OR
A StoryWalk® is a book that has been dismantled, laminated and presented page by page on a trail, bike path, walkway, park, or other outdoor space to promote reading and physical activity. StoryWalks® was created by Anne Ferguson of Montpelier, VT and developed in collaboration with the Vermont Bicycle & Pedestrian Coalition and the Kellogg Hubbard Library. Presently, Vern Patrick is implementing a StoryWalk® on its campus with a homemade set built from metal stakes and yard signs on loan from the Deschutes County Library. On "Wacky Wednesdays," students are treated to literary works and movement on the StoryWalk® at PE time.
As an "exceptional sensory break," Vern Patrick's StoryWalk® resonates with students regardless of age, learning style and reading ability. Students effortlessly spark discussions in the new learning environment outside of the classroom that educators say flow naturally and endlessly.
Vern Patrick would use the Community Builder Award to install a permanent StoryWalk® along the track for the school community to enjoy.
Our schools have been through unparalleled circumstances this year and it is the innovative projects like those described above that will jump-start in-person learning for kids across our region. We hope you will join us in supporting the project you think will make the biggest impact by casting your vote today!
Community Impact StoryTags: Education Kids Banking Smarter Spending
Cast your vote today.
Now through May 4, 2021 you can vote for one of our nine finalists for the Community Builder Award. The winning school, selected by community votes, will receive a $5,000 donation, while four other schools chosen by the OnPoint Prize Selection Committee will each receive $2,000.
Who should receive the $5,000 award?
Cast your vote now
Community Impact Story
OMSI reaches excluded communities to help achieve a more equitable future
The gift of giving—Little Critter Crew wins $5,000 grant in OnPoint's 5th annual social giving campaign
$100,000 in donations for Giving Tuesday
Celebrating 90 years with stories, gratitude and a sweepstakes!
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Then we got the right plane for you.
Owners of the big Discus pakage get a 30% discount!
Dann haben wir das Richtige für dich.
Besitzer des großen Discus Pakets bekommen 30% Rabatt!
No I'm sorry, this is not possible, but you'll get 30% discount when you buy the DiscusK if you already own the big pakage.
CumulusX hat nichts mit Approaching Innstbruck zu tun. Nur in einem Umkreis um Innsbruck hast du die Funktionalität der Vollversion von CumulusX.
was hat es mit dem von dir angesprochenen "grossen Discus-Paket" auf sich; welchen Inhalt hat das Paket, zu welchem Preis und wo kann man es auspacken?
Darin enthalten sind auch einige motorisierte Varianten und vor allem der C4 Endanflugrechner mit vernünftigem Variometer etc.
I was happy to see an aerobatic sailplane offered by aerosoft, but why the Discus K? certainly not a competitive offering, perhaps it was easy to do from the existing discus? I have flown it extensivly on FSX with realism set to high and find it very fragile and unrealistic, i have flown in a real ASH 25 pulling a positive 7-8 Gs, (in the hands of a factory pilot) this one the D-K proportedly sressed to aerobatics breaks. Perhaps you need to tweak the flight characteristics to reflect the real world, not just the design specs.. Why not a Fox or Swift, I would love to see either of these dedicated aerobatic ships.
Not to criticise too much, it is still an excellent sailplane rendition by Aerosoft, make more.
What a cheek to call the Discus K unrealistic, without ANY experience in the real thing!
Within its limits it is pretty close to the real thing. The limits are from FSX but they are more or less exactly where they were in the real thing. For example the inverted flight wasn't very well. So it is not in FSX.
The Discus K is by far the most realistic aerobatics plane in FSX because FSXs physics are not able to simulate more!
Why no Swift, Fox, Extra??? Because it is NOT possible. Other companies sell such aircraft and tell they fly realistic, but they don't!
While I totally understand your defensive attitude, I don't buy the fact that the discus K is the only aerobatic plane that could in theory be modeled in FSX physics. I am not a soft ware programmer but I am a pilot and airplanes don't behave as does the discus K. The compaqrison to an ASH 25? yes your right strange and telling, in the hands of an expert experienced professional pilot sailplanes don't suddenly break at design limit thresholds. Blame it on FSX physics if you want.
As a plus I do appreciate that you have to watch the g meter carefully in order to keep the plane within its modest limits. Give some reasons why a Fox or Swift could not be modeled in FSX. I am interested in knowing more.
Read Joachims post carefully again. He only says that the Discus K is the most realistic aerobaticsplans currently modelled in FSX, not that no others could be done. He only says that others can't be done more realisticly.
The reason why no Fox or Swift can be modeled is that there wouldn't be enough customers to buy it to earn enough money to make to project economicly sensefull. The Discus K was chosen because parts of the old Discus X project could have been used in the K model, therefore reducing development costs and making it an economic success. A Fox or similar planes would need to be made from ground up and therefore eat up more money in development. As they could only be sold at the same price than the Discus K (Customers wouldn't buy it anymore if it was more expensive) the developer would get less money from the project. That's the reason.
in fact I was referring to the technical fact. FSX is not able to simulate inverted stalls for example. Hence it is not possible to do inverted spins or snaps. The whole flight model is too sluggish as well. When flying a plane with a higher roll rate, it does not stop immediately when you center the stick. Try this in the Extra, a nightmare. The original stoppes immediately and hard, the FSX one (and the "aerobatics planes") add ons slowly stop their roll rate.
The physics are simply not suitable for real aerobatic planes.
THESE are the main reason for not doing a real aerobatics plane for FSX.
I really think a realistic flying Extra or Swift would sell but is simply impossible hence no serious developer would dare to try to develope and sell one.
Why no Swift, Fox, Extra??? Because it is NOT possible.
Thanks for the explanation, I accept the fact of engineering costs making it prohibitive to produce a unique prototype., it makes good business sense. Scwiegler did in fact say what I thought he said , see post asbove i messed up the quote function. I accept the fact that FSX has its limitations, its what it is, but it would be quite possible to model a sailplane with 8G positive and 6G negative tolerances, we accept The limitations of Fsx, piloting technique can overcome the "oversteer" that FSX imposes, why use that as an excuse not to produce high performance sailplanes? It would be very nice to have a more worthy offering from you fellows who obviously have the expertise to accomplish this. Thanks for your further explanations, it is fine, I understand the economics and broached same in my first post. The K is a very realistic plane within its limits, it teaches disipline.
I suppose Alebo and IRIS are serious developers and they have attemted aerobatic planes, But I understand your penchant for extreme accuracy.
As an aircraft designer in the real world, I am not encumbered by simulation mechanics, only aerodynamics, of which FSX seems to have an excess of (too much fluidity). I designed a nice little aerobatic sailplane for X plane, but that is easier for me, as the way it handles aerodynamics seems intuitive. It and other Xplane renditions seem to have a more realistic system for aerobatic control response , the planes stop precisly on imput. Too bad the overall package of X plane seems to lag in realism.
Thank you for your fine models and high standards.
Is flutter in some way modelled into the Discus X, K?
I do not have FSX installed anymore, but I did buy the DiscusX, which I consider by far the best glider model for fsx!
A nice 2013 to you all!!!!
I believe that, given the limitations imposed by the uderlying platform (fsx) the discusx is the best soaring experience I had in fsx!
It does not model flutter, as I tested by taking it to different altitudes and taking it to the limits until overstress pumps in.


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