What Happens When Everyone Is Supposed to Believe the Same

However small, however modest, however tentative this may be, it will perpetually give rise to contests and divisions.  And this quest of a common denominator in contrasting convictions can develop nothing but intellectual cowardice and mediocrity, a weakening of minds and a betrayal of the rights of truth. (Jaques Maritain, True Humanism, Bles, 1938, p. 167)

Fading Glory

Next month, Lord willing, I start the second year of my PhD adventure in Computational Engineering.  “Adventure” is a good way to describe it, especially in my superannuated state.  In the midst of getting the coursework started last fall (yes, Europeans, sad to say we have coursework with PhD programmes) we had a “freshman” orientation.

One of the things my advisor oriented us about were the substantial computer capabilities that the SimCentre has.  “Substantial” is a relative term; at one point we were in the top 500 computers in the world for power, but he chronicled our falling ranking which eventually left us “off the charts”.  In spite of that the SimCentre continues to tout its capabilities and the rigour of its curriculum, and I can attest to the latter.

In my desperate attempt to re-activate brain cells long dormant (or activate a few that had never seen front-line service) I dug back and discovered that most of the “basics” of the trade were in place when I was an undergraduate in the 1970’s.  Part of that process was discovering, for example, that one of the books I found most useful for projects like this had as a co-author one of my undergraduate professors!  The biggest real change–and the one that drove just about everything else–was the rapid expansion of computer power from then until now.  What has happened to the SimCentre was not that the computer cluster there had deteriorated; it has just not kept up with the ever-larger supercomputers coming on-line out there.  Such is the challenge the institution faces.

One of my advisor’s favourite expressions is “in all its glory”, but in this case what we have is a case of “fading glory”.  The SimCentre isn’t the only person or institution to face this problem:

If the system of religion which involved Death, embodied in a written Law and engraved on stones, began amid such glory, that the Israelites were unable to gaze at the face of Moses on account of its glory, though it was but a passing glory, Will not the religion that confers the Spirit have still greater glory? For, if there was a glory in the religion that involved condemnation, far greater is the glory of the religion that confers righteousness! Indeed, that which then had glory has lost its glory, because of the glory which surpasses it. And, if that which was to pass away was attended with glory, far more will that which is to endure be surrounded with glory! With such a hope as this, we speak with all plainness; Unlike Moses, who covered his face with a veil, to prevent the Israelites from gazing at the disappearance of what was passing away.  (2 Corinthians 3:7-13)

When Moses came down from Mt. Sinai, having received the law directly from God (a truth that Muslims, along with Christians and Jews, affirm) he wore a veil:

Moses came down from Mount Sinai, carrying the two tablets with God’s words on them. His face was shining from speaking with the LORD, but he didn’t know it. When Aaron and all the Israelites looked at Moses and saw his face shining, they were afraid to come near him. Moses called to them, so Aaron and all the leaders of the community came back to him. Then Moses spoke to them. After that, all the other Israelites came near him, and he commanded them to do everything the LORD told him on Mount Sinai. When Moses finished speaking to them, he put a veil over his face. (Exodus 34:29-33)

Moses’ fellow Israelites were afraid to even come near him on account of the glory of God on his face, thus the veil.  And Paul dutifully replicates this reasoning in 2 Corinthians 3:7.  But Paul throws in another reason Moses needed to wear a veil: to hide the fact that, like the SimCentre’s computers,  the glory was fading!  That is what I call a “2 Corinthians 3” problem, and although Paul applies it to the Jewish law, it has broader application as well.

Today, in spite of our “flattened” society, we see people and institutions lifted up and presented as glorious.  This is especially clear in the “messianic” streak that has entered our political life, but it’s also clear in the adulation given to celebrities, corporations and their products.  It would behove us, however, to take a critical look at such things with one question: are we looking at real, enduring glory, or is this just another example of fading glory which is being covered up with hype?

As Paul and other New Testament authors note, the purpose of Jesus Christ coming to the earth was to solve the problem of fading glory, and specifically of a system that could not produce an entire solution to the problem of our sins and imperfections.   Once that happens within ourselves, we can stop living behind the hype of whatever fading glory we’ve tried to hide behind and live in the truth.

‘Yet, whenever a man turns to the Lord, the veil is removed.’ And the ‘Lord’ is the Spirit, and, where the Spirit of the Lord is, there is freedom. And all of us, with faces from which the veil is lifted, seeing, as if reflected in a mirror, the glory of the Lord, are being transformed into his likeness, from glory to glory, as it is given by the Lord, the Spirit. (2 Corinthians 3:16-18)

If you want glory that lasts, take a look at this

Frederick Gere and Milton Williams: The Winds of God

Century 27269 (1965)

For those of you who survived the 1960’s folk Mass scene, some questions: did you ever wonder how “Kumbayah” became emblematic of people sitting around, holding hands?  Or why Michael had to row the boat ashore when technology of the time had outboard motors (as depicted on the right)? Or why, with so much rich music from the black church, we always fell back on “Were You There”?  This “folk Mass” (more about the quotes later) may be the answer to some of these questions; it is certainly a pioneering work in the genre.

It originated in the San Francisco bay area, at St. Paul’s Episcopal Church in Burlingame.  Frederick Gere was an Episcopal minister who had worked at the University of California at Berkeley during a very turbulent time.  It became obvious to him (and many others) that, to reach the generation coming up, more contemporary sounding music was in order, and The Winds of God was his and Milton Williams’, a music professor at the California State College at Hayward, answer.

“Contemporary” is a relative term, however; there are obviously many concessions to traditional Episcopal/Anglican hymnody, not the least of which is the presence of the pipe organ on many pieces.  Given the early date of the album, they didn’t have much folk music to work from.  The Mass proper is reasonable, and the rest of the work is cobbled together from black spirituals, one piece in Hebrew, the 1940 Hymnal (including my unfavourite hymn, “They Cast Their Nets”), pop songs such as “Blowin’ in the Wind” and of course “Kum Ba Yah”.

In spite of the disparate origins of the music, the production comes off surprisingly well.  That’s due to the professional quality of the music direction and the production, including Milton Williams’ excellent operatic voice.  The album cover claims that this performance was at Grace Cathedral in San Francisco; the recording is better than most at minimising the echo that creeps into cathedral-type recordings.  It’s clear that this work had a strong influence on many organists and choirmasters in the Episcopal Church as the 1960’s lurched along with crises detonated by the likes of Gere’s bishop, James Pike.

One thing that the albums include is a narrative by Gere about what the folk Mass was all about.  This is very much a product of the era, although, restricted to one track, it is much less intrusive than his fellow Episcopalian Ian Mitchell or Catholic Sister Germaine, who larded their works with music and explanation of most tracks.

Unfortunately events were moving faster than Gere and Williams had either anticipated or were allowed to follow.  Already Peter Scholtes‘ “Missa Bossa Nova” was pushing styles in a more folk/rock direction with his inner city musicians, and fellow Episcopalian Tom Belt would shortly break out of the “high church” mould entirely with God Unlimited.  But this is a nice production that, for all of its conservatism and inconsistency, grows on you, as it did for many back in its day.

The songs:

  1. Who Has Seen the Wind?
  2. The Symbolism of the Wind
  3. Michael, Row Your Boat
  4. Six-Fold Kyries
  5. Kum Ba Yah
  6. The Nicene Creed
  7. They Cast Their Nets
  8. Ovinu Malkeinu
  9. He’s Got the Whole World in His Hands
  10. Sursum Corda
  11. Sanctus
  12. The Lord’s Prayer
  13. Were You There?
  14. In Christ There is No East or West
  15. Amen
  16. Blowin’ in the Wind

Download The Winds of God

For all of our music click here

The Importance of Keeping the Riff-Raff Out

Club memberships have been a part of my family tradition since the Gilded Age.  Although it isn’t one of our more prestigious memberships, my business membership in Sam’s Club is doubtless one of the more useful.  One of the nice perks with such a membership is the ability to get in and get your shopping done before 1000, when they open up the place to the general membership and the check-out lines assume Wal-Mart proportions.  When I’m able to take advantage of this, Palm Beacher that I am, I tell my wife that I’m going to Sam’s “before the riff-raff gets there.”

Such sentiments don’t do anything for my Christian humility (such as it is), but they’re instinctive.  (For NT students, this is a “Romans 7” moment).  Keeping the riff-raff out is Rule #1 for an exclusive community of any kind, and it certainly drove much of Palm Beach’s life when I grew up there.  It drove Bethesda-by-the-Sea Episcopal’s Church’s vestry to eject the ladies’ rummage sale from the church grounds.  It also nearly derailed Palm Beach getting its beloved Publix.

Today, of course, our elites assure us that they are the product of a “meritocracy”, as opposed to the old “WASP inherited” deal.  They also assure us that the world is “flattening” due to social media, so the stratified social structure of days gone by no longer exist.

Such self-congratulation, however is belied by stuff like this:

As I have argued elsewhere, there are two competing models of successful American cities. One encourages a growing population, fosters a middle-class, family-centered lifestyle, and liberally permits new housing. It used to be the norm nationally, and it still predominates in the South and Southwest. The other favors long-term residents, attracts highly productive, work-driven people, focuses on aesthetic amenities, and makes it difficult to build. It prevails on the West Coast, in the Northeast and in picturesque cities such as Boulder, Colorado and Santa Fe, New Mexico. The first model spurs income convergence, the second spurs economic segregation. Both create cities that people find desirable to live in, but they attract different sorts of residents…

Finally, there’s the never-mentioned possibility: that the best-educated, most-affluent, most politically influential Americans like this result. They may wring their hands over inequality, but in everyday life they see segregation as a feature, not a bug. It keeps out fat people with bad taste. Paul Krugman may wax nostalgic about a childhood spent in the suburbs where plumbers and middle managers lived side by side. But I doubt that many of his fervent fans would really want to live there. If so, they might try Texas.

Keeping out fat people with bad taste…now that’s the Palm Beach way!  By using the regulatory process to restrict land use, our “flattened” elites have managed to create exclusive enclaves for themselves from which to rule the rest of us, all the while presiding over a widening gap between themselves and the rest of the country.

Palm Beach itself is a good example of this in action.  The Bloomberg article noted the high cost of living in élite places, and that of course starts with housing.  We sold our Palm Beach house forty years ago next month.  A little figuring and research tells me that, even in the current market, the value of the house and land (mostly the latter) has risen about ten times the rate of inflation in the intervening years.  Unlike Paul Krugman, we didn’t live down the street from the plumber, but today I couldn’t even live on my street.

To build anything on the island is next to impossible, although it’s certainly possible to replace a home under the right conditions.  (One thing that makes that simpler is the fact that many homes in Palm Beach aren’t visible from the street because of the foliage).  The island is larded with places on the National Register of Historical Places, including my home church.

With commercial development things are even more complicated.  The aforementioned Publix found the only way to rebuild this island institution was to go around the dreaded Architectural Commission (ARCOM) and come to the Town Council to get the approvals it needed.  Testa’s Restaurant, a more ancient institution where my grandparents dined, has flirted with financial disaster in the process of attempting to get its property redeveloped.  The rent structure of the place has even driven out long-running (55 years) places like Hamburger Heaven.

The result from the island standpoint is that, while locals wring their hands over the problems of redeveloping places like the Royal Poinciana Plaza (where I used to go to Abercrombie & Fitch before they went mass market) more and more retail leaves the island, forcing the residents (especially those who actually live there and not the pied à terre set) to go off the island and mix with the riff raff.  It’s dreadful.

Although Palm Beach is an out-sized example of just about everything, exclusive communities, heavily regulated by land use restrictions and populated by people who can manipulate and outlast such an environment for their own benefit, dot our country and house those who make decisions for everyone.  Even with the pervasive influence of social media, once you achieve geographic segregation, all other kinds follow.

The thing that bothers me more than anything else about this is that our elites can develop a “keep the riff-raff out” driven agenda and still proclaim the society that results as the most just and fair society in human history, denigrating all that has gone before it.  The capacity of human beings for hypocrisy, self-righteousness and blindness to same never ceases to amaze.

And as for those Evangelical Christians on the outside?  One of the big problems Evangelicals face is that their whole life view and structure came up in an open structure.  That structure is still very much alive but, short of splitting the country up, I think that the centralisation of power and money will continue apace, progressively shrinking it.  But I still believe that, with a little paradigm shift, we can adjust.  As I pointed out in the About page:

It used to be, for those of us who happen to live in the United States, that exclusivism and snobbery were the kiss of death in an open and egalitarian society.  But times have changed.  Today we live in a society where the road to the top is well marked educationally and credentially, and becoming more so all of the time.  We even put into the highest office someone who can justifiably be characterised as an elitist snob.

However, for a Palm Beacher such as myself, Jesus’ claims of exclusivity were never a put-off.  In fact, in a place where exclusive clubs and other élite organisations (including churches like the one pictured here) were a natural part of the landscape, the thought of God himself proclaiming an eternal exclusive club was a major part of the appeal.  And having a very well-defined road to joining that club was a natural too.

But the best news of all was this: the membership of this club is open to all who will follow the Master, and he has already paid the dues.

If you’re interested in joining this club, click here for more information

It’s Always Something with a House of Worship, and That Includes a Synagogue

Temple Emanu-El and the Town of Palm Beach go to the mat on another absurd “code violation”:

Temple Emanu-El must reduce the number of seats in its auditorium by more than half, or obtain Town Council approval for the 542 seats that it has.

In a July 12 code violation notice, the town informed the synagogue that it must cut the auditorium capacity to 245 permanent seats, the number approved by the council in 1979, to comply with the town code.

Town officials say they recently learned the synagogue, at 190 N. County Road, more than doubled its seating capacity during an expansion in the early 1990s, but apparently didn’t obtain council approval for the additional seating.

Christians like to think that they are the only targets of code and zoning attacks.  But this is not always the case, and it doesn’t always involve house meetings, either.  Temple Emanu-el isn’t a “new kid on the block” by any stretch of the imagination; they were organised in 1963 (their first service was at Bethesda) and have been in their current location since 1974.

When they expanded, they received a building permit at the start and an occupancy certificate at the end.  One would like to think that the Town of Palm Beach could afford a building inspector who could tell a 245 seat auditorium from a 542 seat one.  So the Town hasn’t exactly been in the dark about this place.  But, as Gilda Radner would say, it’s always something…

I trust that Christians will express their support to the Temple in defending their position.  And it is my fervent hope that God’s Chosen People, having retained suitable counsel, will let the Town have it on this one.

They Love to Feel the Whip

One of the most ill-starred love stories of the last century was that of Tsar Nicholas II of Russia and Tsarina Alexandra.  Neither was equipped by temperament or upbringing to lead an absolute monarchy into World War I.  Nicholas was a weak-willed heir thrust into his role prematurely and unwillingly by the death of his father.  Alexandra was caught up by her son’s haemophilia and the apparent ability of the “monk” Gregory Rasputin–one of the most infamous religious charlatans of all time–to relieve its symptoms.

With the onset of war, their weaknesses–especially his–became apparent to friend and foe alike.  Alexandra, influenced by Rasputin, played favourites with ministers and generals, making an already dicey leadership situation impossible.  She was suspicious of any move towards democratisation or and devolution of power from the throne.  As 1916 came to an end, she exhorted her weak-willed husband to follow the advice of our “Friend” (Rasputin) and the following:

Russia loves to feel the whip…Crush them all under you. (R. Bruce Lincoln, Passage Through Armageddon, p. 282)

“Loves” is probably too strong; let’s just say that, from Ivan the Terrible onward, Russia was good at feeling the whip.  Submission to absolute authority was the rule there.  In this instance, after two and a half years of war, Russia tired of the whip and the deprivations that came with it: Nicholas was forced to abdicate.  Unfortunately years of autocracy rendered Russia unable to make that magical transformation to full democracy our elites think is normal.  After a summer of Alexander Kerensky and an experiment with representative government of sorts, Russia ended up with Lenin and the Bolsheviks, who swapped the whip for the firing squad.  Some of their first victims were Nicholas, Alexandra and their children, gunned down at Yekterinburg.

Evidently the love to feel the whip didn’t die with the Russian monarchy, as has been demonstrated in the overwhelming success of Fifty Shades of Grey.  It’s overwhelmed just about everyone: the author, the publisher, amazon.com…and Evangelical Christianity, which was suitably blind-sided by its success.  Having spent years battling pornographic addiction with men, everyone woke up and groaned at the proposition that we now had to deal with it with women, too.

The response has been interesting, to say the least.  The responsible, if somewhat academic, comeback comes from Dale Coulter.  Not so responsible is that of Jared Wilson, who in quoting Douglas Wilson has started a fire-storm amongst Christian feminists.

Although readers of this blog will take any claim of mine to “cut to the chase” on a issue with a grain of salt, I will try to set out simply what I think are the core problems with this situation.

The changes in the roles of men and women have probably been the largest social change of the last half century, the LGBT community’s attempt to co-opt that notwithstanding.  In Main Line Christianity the rationale behind the whole move towards “equality” of roles (including ministerial ones) was imported from secular feminism, a trend we also see these days in Evangelical Christianity.

The complicating factor in all of this is the place of sexual freedom.  Early feminists, seeing the most straightforward way of eliminating patriarchy is to get the patriarchs out of their lives, demanded a separation.  The main legacy of that these days is the whole corpus of sexual harassment legislation which has become part of the legal landscape/maze that employers deal with on a daily basis.

This separation has proven unsustainable in practice.  Our society is good at marginalising “politically incorrect” views, but sometimes it’s hard to keep them from bubbling to the surface.  The first major sign of trouble was the Monica Lewinsky mess.  Bill Clinton should have been in the feminists’ cross-hairs over that, but they chose political expediency over principle.  The Republicans should have found it easy to impeach him over this and matters related to Clinton’s serial womanising, but his personal popularity and general attitudes blunted that.

Now we have a situation where a woman who loves to feel the whip (or at least finds it expedient to do so) is the heroine of millions of women.  The surprise isn’t that stuff like this is going on; the surprise is the extent to which it has enthralled our society in general and women in particular.  Is this what we’d like to show during “Take Your Daughter to Work” day?  Is this the kind of message that women really want to collectively send to men?  I suspect that the last question is less important to younger people than it is to Boomers and their seniors.

Maybe it should be.  Today’s roles, and the way men and women interact, are largely buttressed by an elaborate system of “rights” which protect women from much of what they had to go through in the past.  As long as the powers that be stuck with principle, these were reliable.  But our whole judicial system and the executive power to back it up is shifting towards a more “outcome-based” system, where the system’s desired results drive the system rather than individual protection. As wealth and power in our system concomitantly centralise, such “rights” become more problematic.  Without these we, like the Russians of old, will feel the whip whether we love it or not.

And, as we know all too well, yesterday’s fantasy can become today’s reality pretty quickly…

As for Christianity, we simply need to make up our minds that we will do things for Biblical reasons rather than to mould ourselves to a societal ideal construct, whether that construct be past, present, or future.  The alternative is to become pointless.

Error Function for an Hermite Polynomial

Our goal is to demonstrate that, for the Hermite polynomial

H_{2n+1}(x)=\sum_{j=0}^{n}f(x_{j})H_{j}(x)+\sum_{j=0}^{n}f'(x_{j})\hat{H}_{j}(x)

where

H_{j}(x)=[1-2(x-x_{j})L'_{j}(x_{j})]L_{j}^{2}(x)

\hat{H}_{j}(x)=(x-x_{j})L_{j}^{2}(x)

the error function is given by the equation

f(x)-H_{2n+1}(x)=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}

where

f\in C^{2n+2}[a,b]

Let us begin by considering a point \hat{x}\in[a,b] where \hat{x}\neq x_{j}, i.e., it is not equal to any of the points on which the interpolant was developed. Since our objective is to determine the error between f(x) and H_{2n+1}(x), because by definition the two are the same at the interpolating points x_{j}, it would be pointless (sorry!) to use one of the interpolation points for \hat{x}.

Now we build a polynomial of degree 2n+2 to describe the error function f(x)-H_{2n+1}(x). This function would interpolate at all x_{j} and additionally \hat{x} for H_{2n+1}(x). This function yields zero error to itself at \hat{x} as an interpolating point. However, by comparing this polynomial at \hat{x} with f(x)-H_{2n+1}(x), we can establish the degree of error. Let us write this polynomial as

H_{2n+1}(x)+\lambda\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}

The constant \lambda is intended to make the interpolant precise at \hat{x}. Let us now state the error of this new interpolant as

\phi\left(x\right)=f\left(x\right)-\left(H_{2n+1}(x)+\lambda\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}\right)=f\left(x\right)-H_{2n+1}(x)-\lambda\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}

Since \hat{x} is an interpolating point, \phi\left(\hat{x}\right)=0. Substituting this into the above and solving for \lambda, we have

\lambda=\frac{f\left(\hat{x}\right)-H_{2n+1}(\hat{x})}{\overset{n}{\underset{j=0}{\prod}}\left(\hat{x}-x_{j}\right)^{2}}

For the other interpolating points, we know that

f\left(x_{j}\right)-H_{2n+1}(x_{j})=0

and, since the Hermite polynomial also interpolates at the first derivative,
f'\left(x_{j}\right)-H'_{2n+1}(x_{j})=0

and finally, obviously,
\overset{n}{\underset{j=0}{\prod}}\left(x_{j}-x_{j}\right)^{2}=0

we can say

\phi\left(x_{j}\right)=f\left(x_{j}\right)-H_{2n+1}(x_{j})-\lambda\overset{n}{\underset{j=0}{\prod}}\left(x_{j}-x_{j}\right)^{2}=0

and

\phi\left(\hat{x}\right)=f\left(\hat{x}\right)-H_{2n+1}(\hat{x})-\lambda\overset{n}{\underset{j=0}{\prod}}\left(\hat{x}-x_{j}\right)^{2}=0

It’s also possible to say that

\phi'\left(x_{j}\right)=f'\left(x_{j}\right)-H'_{2n+1}(x_{j})=0

From this we can determine that \phi\left(x\right) has at least n+2 zeroes (all of the points x_{j} plus the point \hat{x}) in \left[a,b\right]. Likewise we can say that \phi'\left(x\right) has at least n+1 (all of the points x_{j}) zeroes in \left[a,b\right].

At this point we observe the following:

…Rolle’s Theorem states that a continuous curve that intersects the x-axis in two distinct points A\left(a,0\right) and B\left(b,0\right), and has a slope at every point \left(x,y\right) for which a<x<b, must have slope zero at one or more of these latter points. (Tierney, J.A. Calculus and Analytic Geometry. Boston: Allyn and Bacon, 1972, p. 128.)

There is thus at least one zero for each interval; since there are n+1 intervals, we can say from this that \phi'\left(x\right) has at least n+1 zeroes. However, \phi'\left(x\right) also has n+1 zeroes as an interpolant, so \phi'\left(x\right) has a total of 2n+2 zeroes.

Successive differentiation will yield the following

\phi'\left(x\right)\Longrightarrow 2n+2 zeroes
\phi''\left(x\right)\Longrightarrow 2n+1 zeroes
\phi'''\left(x\right)\Longrightarrow 2n zeroes
\vdots
\phi^{\left(2n+2\right)}\Longrightarrow 1 zero

From this we can conclude that, for the one zero of the final derivative

\phi^{\left(2n+2\right)}\left(\eta\left(x\right)\right)=0

where \eta\left(x\right) is the value where the zero exists.

At this derivative, from our previous considerations,

\phi^{\left(2n+2\right)}\left(x\right)=f^{\left(2n+2\right)}\left(x\right)-H_{2n+1}^{\left(2n+2\right)}(x)-\lambda\left(\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}\right)^{\left(2n+2\right)}

It is fair to say that, because of the degree of the polynomial,

H_{2n+1}^{\left(2n+2\right)}(x)\equiv0

The last term could be quite complex to differentiate, but let us
consider the following:

\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}=x^{2n+2}+r\left(x\right)

where r\left(x\right) is a polynomial. Taking the 2n+2 derivative, r\left(x\right) disappears and we are left with

\left(\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2}\right)^{\left(2n+2\right)}=\left(2n+2\right)!

Substituting,

\phi^{\left(2n+2\right)}\left(x\right)=f^{\left(2n+2\right)}\left(x\right)-\lambda\left(2n+2\right)!

Solving,

\lambda=\frac{f^{\left(2n+2\right)}\left(x\right)-\phi^{\left(2n+2\right)}\left(x\right)}{\left(2n+2\right)!}

At the point \eta\left(x\right), \phi^{\left(2n+2\right)}\left(\eta\left(x\right)\right)=0,
and now

\lambda=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}

Recalling

\phi\left(\hat{x}\right)=f\left(\hat{x}\right)-H_{2n+1}(\hat{x})-\lambda\overset{n}{\underset{j=0}{\prod}}\left(\hat{x}-x_{j}\right)^{2}=0

or

f\left(\hat{x}\right)-H_{2n+1}(\hat{x})=\lambda\overset{n}{\underset{j=0}{\prod}}\left(\hat{x}-x_{j}\right)^{2}

we can substitute and achieve our original goal

f\left(\hat{x}\right)-H_{2n+1}(\hat{x})=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}\overset{n}{\underset{j=0}{\prod}}\left(\hat{x}-x_{j}\right)^{2}

Preventing the Anglican Revolt in the Church of God: The Obvious Rationale Behind Agenda Item #18

As I mentioned earlier, the Church of God will convene its General Assembly next week in Orlando.  The agenda is out and posted.

Most church meeting agendas like this are full of arcane items that may be very weighty to those whom they directly impact but of little relevance to everyone else, even for general interest.  However, most agendas have at least one item that breaks that mould.  This go around we have Agenda Item #18, which means it will be taken up at the bottom of both General Council and General Assembly (for my Anglican and other visitors, going into the details about how that works is a long business which I’ll skip).

The agenda item reads as follows:

10. Responsible Use of Social Media
Christians are exhorted by Scripture to speak the truth in love (Ephesians 4:15), to provide things honest in the sight of all persons (Romans 12:17), and to do all things for the edification of others (Romans 15:2). The use of social media (such as MySpace, Facebook, Twitter, blogs, websites, and so forth) by believers should conform to these and other biblical standards.
Church of God ministers, as examples of believers in speech, life, faith, and purity (1 Timothy 4:12), shall at all times agree:
a. To write and post only under their own name.
b. To not attack fellow ministers or members of the Church of God. One may disagree with others, provided the tone is respectful and does not become a personal attack.
c. To not disclose any sensitive, confidential, or financial information about the church, its ministers, or its members, other than what is publicly available.
d. To not post any material that is defamatory, libellous, threatening, harassing, abusive, or embarrassing to any person or entity.
e. To uphold the doctrine of the Church of God by not writing or posting anything contrary to the accepted doctrine of the Church of God.
Failure to follow these guidelines on the use of social media shall result in the offending minister being subject to discipline for unbecoming ministerial conduct.

Let me first make a couple of purely personal observations.

  1. This doesn’t apply to me because I am not ministerially credentialled in the Church of God or anywhere else.  One of these days my church may get its canful of the “Elitist Snob” bloviating the way he does and turn me out, but that’s not what’s being discussed here.
  2. Although people often have trouble figuring out who is behind this blog, I’ve never gone “anonymous” in what I have to say on the Internet.  That’s not because I object to the concept of anonymous presence on the net; there are situations where it is necessary for a variety of reasons.  While going that route wasn’t my choice, I can see why others might justifiably do so.  (That’s my response to (a)).

With that out of the way, let me put forth why I think this should not be adopted.

I’ve had the experience of working in two church “blogospheres” over the last decade or so: the Anglican/Episcopal one first, and then my church’s own, especially during the “Missional Revolt” in 2008-10.  I’ve always contended that the Anglican/Episcopal drama is the most riveting story in contemporary Christianity, and one thing that made it that way was the demonstration of the power of the Internet via websites, blogs and social media.

The Episcopal Church has had a long-term leftward drift punctuated by severe lurches in that direction.  In the 1960’s and 1970’s we ended up with the 1979 BCP and the general dilution of orthodox Christian belief.  What resulted was a severe bleeding of the membership and scattered Anglo-Catholic secession, but the church’s control of the situation remained intact, setting us up for the next lurch: the elevation of V.G. Robinson to the status of bishop.

This go around was different.  Using the Internet, and with the help of the African provinces, Anglicans were able to spread the word that there was an alternative out there to revisionist church.  Although the ACNA and the other Anglican bodies that have come up have their issues, that they exist at all in the strength they have–including old Episcopal congregations and dioceses, with or without the property–is a testament to the power of the Internet and the persistent voices that encamped there.

The Church of God’s situation was entirely different in terms of doctrine and the nature of the dispute, but one result was the same: a direct and effective challenge to the leadership of the church.  Without the blogs and websites of those who sought changes in the financial–and by extension the general–direction of the church, the changes that were mandated in 2008 and subsequently implemented would have never taken place.

Whether this is a net benefit or deficit is something that only time will tell.  There’s no doubt, however, that our leadership would like to prevent a repeat of that.  Much of what’s in the item (especially b-d) sounds very nice.  But why should this be spelled out only for online speech when much of it has more general application for Christians?  In that respect the agenda item is superfluous.

One of these days the ministers and lay people of our church are going to have to “go to the mat” over more than just where the money goes.  In many ways this item is like the Anglican Covenant: it looks great up front, but in the future could be used as a club by revisionists who get control of the apparatus.

I trust that our ordained bishops will see the sense in voting down this item and not force the laity to do it for them.

Note on item (e): That’s another item that many of my Anglican friends probably would take heart in.  But many Church of God ministers–even conservative ones–are very leery at enacting explicit “doctrinal tests” like this.  Their rationale is rooted in the history of modern Pentecost, but the truth is that discussions figuring out what’s orthodox and what’s not often fall to a low level, which would turn explicit adherence clauses like this into a club to be used by the minister with the biggest mouth.  (It’s not that much better elsewhere, if you’re wondering).  How we’re planning to handle the coming assault by the LGBT community is the question we need to be spending time on.  But we’re not.

Stability of Back Substitution for Matrix Systems

The objective is to show that back substitution is backward stable.  Consider the system

Rx=c

where R is an upper triangular m\times m matrix and x,c are m column vectors. For a 3\times3 matrix system, this would look like:

\left[\begin{array}{ccc}  r_{1,1} & r_{1,2} & r_{1,3}\\  0 & r_{2,2} & r_{2,3}\\  0 & 0 & r_{3,3}  \end{array}\right]\left[\begin{array}{c}  x_{1}\\  x_{2}\\  x_{3}  \end{array}\right]=\left[\begin{array}{c}  c_{1}\\  c_{2}\\  c_{3}  \end{array}\right]

Now let us consider the perturbation matrix

\delta R=\left[\begin{array}{ccc}  \delta r_{1,1} & \delta r_{1,2} & \delta r_{1,3}\\  0 & \delta r_{2,2} & \delta r_{2,3}\\  0 & 0 & \delta r_{3,3}  \end{array}\right]

and a computed solution for x

\hat{x}=\left[\begin{array}{c}  \hat{x_{1}}\\  \hat{x_{2}}\\  \hat{x_{3}}  \end{array}\right]

A system is backward stable if, on a computer with standard machine arithmetic, the computed solution satisfies the following:

\left(R+\delta R\right)\hat{x}=c

where

\frac{\Vert\delta R\Vert}{\Vert R\Vert}\leqq\mathcal{O\left(\epsilon_{\mathrm{machine}}\right)}

which is more concisely expressed as

\frac{\mid\delta r_{i,j}\mid}{\mid r_{i,j}\mid}\leq m\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

Let us begin by considering the third row, which purely mathematically solves as follows:

x_{3}=\frac{c_{3}}{r_{3,3}}

Let us introduce a machine perturbation at c as follows, which results in a computed solution

\hat{x_{3}}=\frac{c_{3}\left(1+\epsilon_{3}\right)}{r_{3,3}}

The problem with this formulation is that the perturbation introduced does not match our definition for backward stability, which results in a formulation such as this:

\hat{x_{3}}=\frac{c_{3}}{r_{3,3}+\delta r_{3,3}}

We must thus transform the perturbation in c to one in R. A Taylor series for the perturbation in c allows us to move the perturbation from the numerator to the denominator in this way:

\frac{1}{1+\hat{\epsilon}}=1-\epsilon+\mathcal{O\left(\epsilon^{\mathrm{2}}\right)}

where

\mid\hat{\epsilon}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

This means that we can rewrite our equations for \hat{x_{3}} as

\hat{x_{3}}=\frac{c_{3}}{r_{3,3}\left(1+\hat{\epsilon_{3,3}}\right)}

or

\delta r_{3,3}=\hat{\epsilon_{3,3}}r_{3,3}

Thus

\mid\hat{\epsilon}_{3,3}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

Now that we have solved for \hat{x}_{3}, we can proceed up a row and solve for \hat{x}_{2}. We start by solving for \hat{x}_{2}
as follows:

\hat{x}_{2}=\frac{c_{2}-r_{2,3}\hat{x}_{3}}{r_{2,2}}

Note that we have three operations in this step:

  1. Multiplication in the numerator
  2. Subtraction in the numerator
  3. Division between the numerator and the denominator

This introduces three points of machine error, accounted for thus:

\hat{x}_{2}=\frac{\left(c_{2}-r_{2,3}\hat{x}_{3}\left(1+\epsilon_{2,3}\right)\right)\left(1+\epsilon_{2,1}\right)}{r_{2,2}}\left(1+\epsilon_{2,2}\right)

where, as always,

\mid\epsilon_{2,1}\mid,\mid\epsilon_{2,2}\mid,\mid\epsilon_{2,3}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

Using the Taylor series transformation for two out of the three error expressions,

\hat{x}_{2}=\frac{\left(c_{2}-r_{2,3}\hat{x}_{3}\left(1+\epsilon_{2,3}\right)\right)}{r_{2,2}\left(1+\hat{\epsilon}_{2,2}\right)\left(1+\hat{\epsilon}_{2,1}\right)}

Again,

\mid\hat{\epsilon}_{2,1}\mid,\mid\epsilon_{2,3}\mid,\mid\hat{\epsilon}_{2,2}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

We can combine the two error expressions in the denominator by considering the fact that \hat{\epsilon}^{2}is beyond our considerations, as has been consistently shown. We thus say that

1+2\hat{\epsilon}_{2}=\left(1+\hat{\epsilon}_{2,2}\right)\left(1+\hat{\epsilon}_{2,1}\right),\mid\hat{\epsilon}_{2}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

and substituting

\hat{x}_{2}=\frac{\left(c_{2}-r_{2,3}\hat{x}_{3}\left(1+\epsilon_{2,3}\right)\right)}{r_{2,2}\left(1+2\hat{\epsilon}_{2}\right)}

where

\mid2\hat{\epsilon}_{2}\mid\leq2\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

Equating, as before,

\delta r_{2,3}=\hat{\epsilon_{2,3}}r_{2,3}

and

\delta r_{2,2}=2\hat{\epsilon_{2}}r_{2,2}

Finally we get to the top row. The value for $\hat{x}_{1}$ is given
by

\hat{x}_{1}=\frac{\left(c_{1}-r_{1,2}\hat{x}_{2}-r_{1,3}\hat{x}_{3}\right)}{r_{1,1}}

We have the same possibility for machine error as before, except that we have some additional calculations, each with the possibility of inducing error. Let us rewrite the above as follows:

\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\right)-r_{1,3}\hat{x}_{3}\right)}{r_{1,1}}

which more clearly shows the order of machine operations. The error points can be inserted thus:

\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\left(1+\epsilon_{1,2}\right)\right)\left(1+\epsilon_{1}\right)-r_{1,3}\hat{x}_{3}\left(1+\epsilon_{1,3}\right)\right)\left(1+\epsilon_{0}\right)}{r_{1,1}}\left(1+\epsilon_{1,1}\right)

Performing the Taylor series operation and combining \epsilon_{0} and \epsilon_{1,1}, we have
\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\left(1+\epsilon_{1,2}\right)\right)\left(1+\epsilon_{1}\right)-r_{1,3}\hat{x}_{3}\left(1+\epsilon_{1,3}\right)\right)}{r_{1,1}\left(1+2\hat{\epsilon}_{1,1}\right)}

Multiplying

\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\left(1+\epsilon_{1,2}\right)\right)\left(1+\epsilon_{1}\right)-r_{1,3}\hat{x}_{3}\left(1+\epsilon_{1,3}\right)\frac{1+\epsilon_{1}}{1+\epsilon_{1}}\right)}{r_{1,1}\left(1+2\hat{\epsilon}_{1,1}\right)}

Factoring out in the numerator and applying two more Taylor series transformations, we can say

\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\left(1+\epsilon_{1,2}\right)\right)-r_{1,3}\hat{x}_{3}\left(1+\epsilon_{1,3}\right)\left(1+\hat{\epsilon}_{1}\right)\right)}{r_{1,1}\left(1+2\hat{\epsilon}_{1,1}\right)\left(1+\hat{\epsilon}_{1}\right)}

Combining error terms and eliminating the squared epsilons yields

\hat{x}_{1}=\frac{\left(\left(c_{1}-r_{1,2}\hat{x}_{2}\left(1+\epsilon_{1,2}\right)\right)-r_{1,3}\hat{x}_{3}\left(1+2\hat{\epsilon}_{1,3}\right)\right)}{r_{1,1}\left(1+3\hat{\epsilon}_{1,1}\right)}

Noting as before that

\delta r_{1,1}=3\hat{\epsilon_{1,1}}r_{1,1}

\delta r_{1,2}=\epsilon_{1,2}r_{1,2}

\delta r_{1,3}=\hat{2\epsilon_{1,3}}r_{1,3}

\mid3\hat{\epsilon}_{1,1}\mid\leq3\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

\mid\hat{\epsilon}_{1,2}\mid\leq\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

\mid2\hat{\epsilon}_{1,3}\mid\leq2\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

We are now able to construct the perturbation matrix thus:

\delta R=\left[\begin{array}{ccc}  3\hat{\epsilon_{1,1}}r_{1,1} & \epsilon_{1,2}r_{1,2} & \hat{2\epsilon_{1,3}}r_{1,3}\\  0 & 2\hat{\epsilon_{2}}r_{2,2} & \hat{\epsilon_{2,3}}r_{2,3}\\  0 & 0 & \hat{\epsilon_{3,3}}r_{3,3}  \end{array}\right]

Now we must use this to satisfy the criterion for the backward stability of back substitution:

\frac{\mid\delta r_{i,j}\mid}{\mid r_{i,j}\mid}\leq m\epsilon_{machine}+\mathcal{O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)}

Let us write a matrix in accordance with this criterion and factoring
out the machine epsilon:

\left[\begin{array}{ccc}  \frac{\mid3\hat{\epsilon_{1,1}}r_{1,1}\mid}{\mid r_{1,1}\mid} & \frac{\mid\epsilon_{1,2}r_{1,2}\mid}{\mid r_{1,2}\mid} & \frac{\mid\hat{2\epsilon_{1,3}}r_{1,3}\mid}{\mid r_{1,3}\mid}\\  0 & \frac{\mid2\hat{\epsilon_{2}}r_{2,2}\mid}{\mid r_{2,2}\mid} & \frac{\hat{\mid\epsilon_{2,3}}r_{2,3}\mid}{\mid r_{2,3}\mid}\\  0 & 0 & \frac{\hat{\mid\epsilon_{3,3}}r_{3,3}\mid}{\mid r_{3,3}\mid}  \end{array}\right]\leq\left[\begin{array}{ccc}  3 & 1 & 2\\  0 & 2 & 1\\  0 & 0 & 1  \end{array}\right]\epsilon_{machine}+O\left(\epsilon_{\mathrm{machine}}^{\mathrm{2}}\right)

We can factor the machine epsilon out because we have consistently demonstrated that all of the epsilons are smaller than the machine epsilon, thus the matrix on the right hand side is greater than the one on the left. The greatest single coefficient of the machine epsilon is 3, which equals to the matrix size m. So this criterion is achieved for any and all of the quotients, and thus the backward stability of back substitution is achieved.

Concerning the extension of this pattern to larger matrices, the number of errors–and thus the value of the coefficients in the normalized matrix we wrote at the end–depends upon the number of calculations and thus points of error. These can be discussed as follows:

  • Division: each calculation has one division. The error resulting from this division appears in the main diagonal because the denominator is always the coefficient in the diagonal, thus one error for each diagonal position is a result of the division.
  • Multiplication: each non-diagonal term is associated with a multiplication because we are solving for the \hat{x} associated with the diagonal. So one additional error is added to each non-diagonal element in the upper portion of the matrix for multiplication.
  • Subtraction: there are as many subtractions as there are multiplications, but they end up in the error matrix differently. The situation with subtraction is further complicated by that fact that, in multiplying through the subtractive errors in order to get the error expressions into the denominator, the sum of the subtractive error coefficients ends up to be greater than the total number of error expressions entered into the equations. First, because of the multiplying through to get the Taylor series conversions to the denominator, the number of subtractive errors in the main diagonal (and thus associated with the denominator) is m-i, where i is the row number. For the rest of the matrix, the number of subtractions is j-i-1, where j is the column number.

The sum of any of these errors that apply to a given position in the matrix is the total number of errors in that position, and thus the value of that position in the error matrix.

The critical position is i=j=1. In that position there is one error for division and m-i errors for division. The sum of these is always m, and thus the criterion for stability is maintained as the matrix size increased.

Barack Obama and Small Business: They Wouldn’t Get Very Far Without Us, Either

Things are getting very deep out there:

President Barack Obama addressed supporters in Roanoke, Virginia on Saturday afternoon and took a shot at the business community. President Obama dismissed any credit business owners give themselves for their success:

There are a lot of wealthy, successful Americans who agree with me — because they want to give something back.  They know they didn’t — look, if you’ve been successful, you didn’t get there on your own.  You didn’t get there on your own.  I’m always struck by people who think, well, it must be because I was just so smart.  There are a lot of smart people out there.  It must be because I worked harder than everybody else.  Let me tell you something — there are a whole bunch of hardworking people out there.  (Applause.)

 If you were successful, somebody along the line gave you some help.  There was a great teacher somewhere in your life.  Somebody helped to create this unbelievable American system that we have that allowed you to thrive.  Somebody invested in roads and bridges…

This is of special relevance to me because, just two weeks ago, I posted a piece on the 144-year run of my family business.  (And we were in Chicago for 108 of those).  I have three things to say about this:

  1. Economic interdependence is a given in a capitalist system.  But trying to turn that on its head and say that capitalists deserve no credit for what they did is ridiculous.  Capitalists take risks: some made it and some didn’t.  Just because all that Barack Obama knows about are crony capitalists like those who are bankrolling his campaign doesn’t mean that’s all there is.
  2. The roads, bridges, etc. out there were very much a market for our company, and we know that.  But almost all the construction equipment, including ours, came from private companies.  Without construction equipment manufacturers DOT’s and the military would be in serious trouble, and they know it.
  3. What government gives with the money it invests in productivity enhancing activities such as infrastructure it can take away through thoughtless regulations and punitive–and often political–actions against corporations and their owners.

I’d also throw in a technical point: one of Marxism’s central tenets is that capitalists are immoral by exploiting their workers’ surplus value. But that exploitation is the vehicle by which capitalists are rewarded for the risks they take in a given enterprise, assuming it is successful (which it often isn’t).  A society which prohibits the exploitation of surplus value ends up not producing any and impoverishes itself, which is exactly what the Soviet Union did.  Obama, by demonising capitalism in this way, is setting us up for this result as well.

I’ve always felt that Barack Obama had a special animus for small businesses and small business people, and this is an expression of that animus.  It’s not restricted to Obama, either; it pervades our elites, both in government and in the “private” sector, too.

Like I said, it’s getting deep out there…