At the time we went, Iceland was one of the few countries in the world that allowed vaccinated foreign visitors. All that was needed was proof of vaccination – not even a pre-entry COVID test. The fact that most people were vaccinated meant that most people were not wearing masks in the city, but this was before even the Delta variant – it seems like so long ago now.

I had been itching to go travel somewhere since the pandemic began, and this was a very welcome nudge towards normalcy. Flying with Icelandair was the same as before: the seat and leg room is pretty comfortable, but there’s no food included with the airfare. The only difference is the extra bureacracy related to COVID, which increased how long it took each person to get through immigration, resulting in long lines after the plane. It didn’t help that we arrived so early in the morning that the food places weren’t even open yet and we had to wait in line while hungry. Eventually, we picked up the rental car and drove into Reykjavik for breakfast. We took the day easy by just exploring Reykjavik on foot, and took a nap in the afternoon when the jet lag caught up to us.

That night, we had probably the most memorable meal we’ve ever had at Tapas Barinn. The great thing about tapas is that as tourists, we get to try a little bit of everything as opposed to ordering entrées.

I even got to try some Flóki single malt whiskey. The smokiness from the whiskey actually comes from the barley smoked using sheep dung, a traditional cooking method. I also got to try some more brennivin like my last trip to Iceland – it’s essentially a grain alcohol (40%) and flavoured with caraway seeds, which gives it a sweeter flavour than something like vodka.

I had visited Iceland a couple years ago, but it was in the winter where there was only 4 hours of light per day, and even then the sky looked like dusk.

This time, travelling there in the summer, I didn’t see the night sky the entire time I was there. I could more clearly see and appreciate the nature around me, like the endless fields of moss growing on the rocks, and sheep freely roaming the fields.

It is thought that the Icelandic sheep were introduced to Iceland by the Viking settlers. Farmers allow them to freely roam around in the summer months, and as winter comes, they round them all back up again.

Like my previous trip to Iceland, we travelled along the Ring Road to see the waterfalls, and the volcanic beaches.

This is what the view at Seljalandsfoss looks like in the summer versus the view I saw in the winter. I had no idea Iceland could be so green. In the summer, people can even walk behind the waterfall, between the rock and the falls itself, but in the winter, the trail is closed because of ice.

Skógafoss looks about the same between summer and winter, except the vegetation is greener in summer.

A kitty even came out to hang out with us! As we were walking, the cat just walked up to us and stopped, expecting pets as a payment to pass through. After that, off it went.

Unlike travelling to Iceland in the winter, there were no restrictions (4WD) to drive up to Dyrhólaey in the summer. As we drove up, I could see why. The road was a set of steep switchbacks on gravel roads that were only a little more than two cars width – and there were no guardrails.

From the top, one can see the endless mountains, or watch the waves crash against the beaches of volanic sand.

The next day, we took a Superjeep tour to see ice caves on an active volcano known as Katla. The reason why it’s called Superjeep is because it can deflate its tires (for icy roads), and inflate them back up again depending on the terrain, but otherwise it looks like your typical Ford Excursion but with big tires.

The ice caves were located a ways in from the glacier. In summer, a lot of the ice had melted, revealing the volanic sand and rock underneath.

Only just now as I was writing this, did I realize we were travelling on a literal volcano, but I suppose it’s safe.

As we headed up the volcano, the lush, green vegetation quickly disappeared, revealing a rocky landscape with a mix of ice and black volcanic rock. Grey clouds covered the sky and it began to rain.

Eventually, the tour guide parked the truck, and we began our volcano hike. He brought his ice pick with him and used it several times to cut steps in the ice.

Despite the summer season, the temperature still felt like a mild winter. But it still meant the ice caves would not be as fully developed as in the winter season, and this could clearly be seen when comparing the ice caves from this trip versus the previous one. In fact, the winter ice cave was really more of a tunnel, while the picture of the summer ice cave was taken from the end of the cave with the camera pointing out.

The ice on top of the volcanic rock looked so alien that several movies used this place as a location for another planet. Rogue One was filmed in the area for the opening scene where young Jyn Erso hides from the stormtroopers. The movie doesn’t show the whole cave, but the similiarity is clear. If the movie had shown the whole cave, one would see why people call it the “Yoda Cave”.

It was nice staying in the cave for a bit as a brief respite from the rain. We made the mistake of wearing jeans for a multi-hour hike in the rain, and our wet clothes just made us even colder. We also got the chance to chat a bit with others on the tour. We met an orthodonist who practised in Greece by the name of Thanos (I wonder if he was aware of the MCU), and another who claimed to be a “voodoo priestess”, whatever that means.

The next morning, the plan was whale and puffin watching. It turned out to be an incredibly bad idea because I apparently get seasick, and didn’t get to see a lot. Even those who did see whales could really only see a small fin barely capturable on camera. Only a small percentage of people get to see the whale flying out of the water and crashing back down. Good thing the puffin watching was on a separate tour so I got to recover from the seasickness, but we decided not to go and instead gave the tickets away to another group that we overheard looking for tickets to the same tour.

Instead, we went to go get some Icelandic hotdogs for lunch. The key things that set it apart are the condiments, and the sausage. In addition to ketchup, and sweet mustard, they also add “remoulade”, similar to mayo with added herbs. The sausage uses lamb, as opposed to chicken/pork/beef.

Next, we headed to the Reykjanes Peninsula and the Blue Lagoon. We reached the tip of the peninsula and stood on the cliff watching the waves crash against the shore. If there was such a thing as the end of the earth, this was it. Despite strong gusts of wind, thousands of birds nested on the side of the cliff. From far away they look like white specks, but one can see them clearly with a good set of binoculars.

Next, we went for a relaxing soak at the Blue Lagoon, a man-made spa fed by water used by a nearby geothermal plant.

The water forms a milky blue colour from the high silica mud, and is supposedly good for the skin. They even sell super overpriced skincare products at the gift shop.

The area is beautiful with all the steam slowly rising from the hot springs with the light cool breeze. It would have felt like a whipping biting wind if not for the hot spring. You just submerge yourself until you feel not too hot and not too cold, and then relax with their cleansing silica face masks and a drink.

After leaving Blue Lagoon, we’d learned there was an active eruption happening at the nearby Fagradalsfjall volcano, and was happening slow enough such that it was OK to go see it.

A short walk from the parking lot led us to the lava flow. The trail continued upstream of the lava flow to the mouth of the volcano.

There were warnings for people to not walk too close to the lava flow because of risks of toxic fumes, but many did it anyway (as did we). Many curious hikers threw rocks on it just to find out that the lava flow had already hardened, even though there were still visible fumes coming out.

The hike turned out to be really long, and we weren’t really prepared to do a multi-hour hike at 20:00 in the cold, wind, and rain. As we reluctantly turned around, we saw people pointing the other and when we looked, we managed to catch a glimpse of the volcano spewing lava from afar. It makes sense now why people call it the “land of ice and fire”

It was surprising to see how watery the lava looked. I expected molten rock to be more viscous. The lava just spewed out in a splash. I felt super lucky to have seen such a sight.

As we walked back, I noticed these hexagonal patterns in the ground. Maybe this is how basalt columns form? From what I’ve read, under certain conditions, the cooling magma can cool in a way that forms these hexagonal shapes.

By the time we reached the parking lot, we were wet and cold (seemed to be a recurring theme by this point). Luckily there was a food truck at the parking lot selling hot soup.

The next morning before heading out to the Golden Circle, we started the day off with a nice brunch at Sandholt.

The unexpected star of the breakfast was the bread. As an amateur baker myself, this is the bread I aspire to make – it’s soft, light and airy with a crisp crust.

Next step along the Golden Circle was Geysir. The geothermal heat creates pressure underground causing the water to explode upward every few minutes. The entire area is filled with geothermal pools, some of which are so hot they’re bubbling from the heat and pressure.

It’s cool just to just stand in anticipation of the eruption. From looking at it, there’s no indication that it’s going to explode at all, and then it just does. At one point, the wind carried the water toward us and got us all wet, but luckily by the time the water hits the ground, it’s cooled down.

From the Geysir, we drove to Gullfoss. Unlike other waterfalls, this one has a unique feature where the water does a zig zag before hitting the bottom.

There were attempts in the past to build a hydroelectric plant here, but those efforts have not succeeded, and since then, the waterfall is now under government protection.

The last destination of the Golden Circle was Kerið crater. It’s believed that the crater formed through volcanic activity. Once the magma chamber emptied, it collapsed, forming the crater. The accumulation of water is not formed by rainfall, but is actually ground water.

On our last day before we left, we got the chance to try more Icelandic dishes. This time we had some lamb soup, dried fish, fermented shark, and some Einstök white ale, a local craft brew.

It’s not often that I would rave about soup, but I liked it so much I’ve been trying to recreate the flavour at home. It’s savoury soup with vegetables, lamb, and herbs/spices. I can’t say I was a huge fan of the dried fish and fermented shark though. The dried fish is served with butter and its flavour reminds me of super tough beef jerky with a light fish flavour, while the fermented shark has a texture of a ham with a soury tang. Perhaps it’s an acquired taste. The beer was great though.

That’s pretty much it for Iceland. We got the chance to try all sorts of local delicacies, and see a piece of nature we wouldn’t normally see in our part of the world. At the airport duty free, I even got to pick up some Brennivin, and a bottle of the Flóki sheep dung whiskey!

See more pictures of my trip to Iceland here

Cologne (Köln)

Last time I went to Cologne, I celebrated the Carnival and didn’t really see or do anything else. This time, I spent a half day checking out the chocolate museum and have a few beers.

The museum chronicles the history of chocolate, from the cultivation of the cocoa bean, to the industrial machines that produce the chocolate. Visitors can even order their own custom chocolate bar! They pick the kind of chocolate (white, dark, etc.) and can choose a multitude of condiments too.

I honestly was more impressed with the industrial chocolate making machinery more than the chocolate itself. As a graduate of mechatronics engineering, I can certainly appreciate the engineering behind these machines. The movements of each component is precisely controlled to perfectly pick up and drop pieces of chocolate or to pour liquid chocolate into molds, and the subsequently release them.

Afterward, we met up with a friend of mine who decided to move to Germany from America. He was so interested in Germany that he earned his C1 level in the Goethe Insitut examinations (it’s one level below native speaker), and when talking to him, I could always tell his future was in Germany. He found a job working at Deloitte and his German is so good that he doesn’t even speak English at work! Even I can’t do that with my German.

Next up: the bar. You can’t go to Cologne and not try the local brew, Kölsch. Germans outside of Cologne assert that Kölsch shouldn’t be considered a real beer, and honestly, I find it to be just okay. It’s not great and not terrible. A unique part of the Cologne pub culture is that if the waiter sees you with an empty glass, he will automatically serve you another one, regardless of whether you want one or not. The sign that you’re done is to put a coaster over the glass.

We walked around town a bit and went to probably the biggest landmark in town: the Cologne Cathedral. This Cathedral is the second tallest church in Europe (the first is in Ulm, also in Germany), with a height of 157m. It’s hard to imagine the size, but most of the buildings in Cologne aren’t very tall so this building appears to tower over the skyline.

Düsseldorf

Next, we drove to Düsseldorf. All we did there was walk a bit around town, but the streets just kind of look like other cities nearby, like Cologne, except Cologne has a giant cathedral to look at. We did come across a small celebration in the street though. I’m not completely sure, but I think it’s for the massive Carnival celebration that was gonna happen in late February.

The fun started later that night though. We were looking to get some dinner and went to a local pub. It was Friday night and the place was packed – so you can be pretty sure this place is good. Normally, you’d just walk in and sit at any free table – there’s nobody to show you to your seat like in America. There was nothing free anywhere.

A group of Germans signaled to us that we could join them and they’d make space for us. I politely told them that it was okay and we could wait, but they started shifting over so we ended up joining them. We chatted with them over drinks and they turned out to be super fun people!

A older couple were having beers with their son in law and his friend until we joined (they were drunk already when we showed up). They were so happy to chat with us and kept telling us how awesome we were for visiting Düsseldorf. One guy kept telling us in his German accent how “North Rhine-Westphalia is the most open-minded place in all of Germany”, how he grew up in Bavaria near Allgäu, but can never go back there because Düsseldorf is just so amazing, and that we should live here permanently.

Germans when they’re drunk, really like speaking English. One of them kept getting stuck on words and he’d ask their friends what the word was in English. I’d translate the words for him each time and eventually one of their friends literally just told him, “don’t ask us, ask him! He knows all the words!” They were super impressed by my German and it’s a pretty great feeling to know that my German is good enough to have these conversations with locals.

My friend on the other hand, could more or less understand what was being said, even though he couldn’t speak the words. In fact, he had a bit of trouble with his pronunciation, to the point that not a single server in any pub we visited in Germany understood him when he tried to order a Radler. This bar was no exception either, and I couldn’t help myself but laugh at his misfortune

As we left, we jokingly said we’d meet again at the same place and same time to have beers together again. I gotta say that most Germans I’ve met have been pretty cool.

That’s all for now!

• Cologne
• Düsseldorf
  

The Hague (Den Haag)

The first stop is The Hague in the Netherlands. It’s the home of the International Court of Justice (pictured above). This is the international court that resolves international disputes that the UN is involved in. It was involved in well known cases such as the International Criminal Tribunal for Rwanda to judge the perpetrators of the Rwandan genocide, and the International Criminal Tribunal for the former Yugoslavia to judge the perpetrators of crimes committed during the internal struggles in Yugoslavia that eventually led to its dissolution.

For lunch, we searched around for something to eat and we could try some herring since that’s a popular dish in the Netherlands. We found a street food stall near us not far from where we were and proceeded to go inside and order. One problem though… the entire menu was in Dutch and there was no English to be found anywhere. We stood there looking confused for a moment, before I piped up and simply asked the staff (knowing that the Netherlands has a very high degree of English literacy) what we should order. He simply replied, “herring sandwich!”. It’s raw herring preserved in a brine in a sandwich along with condiments such as raw onions and pickles. It’s got a nice fishy freshness to it (like sashimi), but a saltier and more sour flavour because of the brine. This is going to be one of those foods that I’ll crave once I leave and can’t get anywhere else.

Another cool place to visit in The Hague is the Madurodam – it’s a park built purely on minatures. Imagine famous landmarks in the Netherlands, like the canals of Amsterdam, old town city squares, and palaces, but in miniature form! As a fan of Lego, I can definitely appreciate looking at miniatures!

Rotterdam

Next, we drove to Rotterdam in the Netherlands and stopped by the Market Hall for more food (I’m always down to eat). This time, we had yet another Dutch speciality: the stroopwafel. It’s hard to say no to two thin crispy wafers with gooey caramel in between.

And because I really enjoy eating protein, I couldn’t resist trying some cured meat. I think this is more of a Spanish thing as opposed to Dutch thing though. I found the flavour way too salty for my tastes (and I use a lot of salt when cooking), but I’m not sure what it’s supposed to taste like.

The market hall also has a nice selection of spices – kind of apt considering their colonial past trading spice and other goods in the Dutch East Indies.

And of course cheese! It’s so nice to see all the wheels of cheese, in all these different colours and sizes.

Baarle-Hertog / Baarle-Nassau

After leaving Rotterdam, we headed towards Belgium – well kind of. The cool thing about Baarle-Hertog / Baarle-Nassau is that a border between Belgium and the Netherlands cuts right through the town in seemingly random spots. You could just be walking down the street along the main avenue of the town and randomly end up in the Netherlands, then in Belgium, and then back again.

On our way out of the town, we noticed lane lines that we hadn’t seen before in North America. At first, we didn’t know why the lines were painted like that, but upon closer analysis, the road is simply a road shared by both cars and bicycles and the red section shows where the bikes should be, and the grey area is the section where cars should be when passing a bicycle. By allocating a space for the car to pass, the cars don’t get too close to the cyclists, thus reducing the risk of an accident. How cool!

Antwerp

Soon, we were fully in Belgium in the city of Antwerp and took a stroll in the old town square. The buildings here look pretty Belgian because of the way the roof inclines – it’s as if there are little steps as opposed to a smooth incline (so-called stepped gables).

This concludes our successful first day of the trip! On day two, we will travel to Oosterscheldekering, Ypres, and Dunkirk.

Photos

• The Hague
• Rotterdam
• Baarle-Hertog
• Antwerp

Print Volume

This is the amount volume that the printer is able to print in.  Generally, the measurement is given by the three dimensions of a rectangular prism.  The thing to watch out for here is that just because a printer has a large print volume, it does not mean the printer can reliably print in all of it.  Say the design calls for a thin flat plate.  If the plate is too wide, then the edges of the plate will peel/warp during the print and the flat shape that was intended will be slightly curved.  To visualize what warping is, imagine a flat plate lying on the print bed as it is being printed.  Warping is when the corners of the plate start peeling off the base during a print.  Warping will be explained in a little more detail later on.

Resolution

This describes the accuracy of the nozzle’s positioning.  It is usually measured in microns or inches.  Clearly a smaller value means that the printer will be more accurate in its position.

Layer Thickness

Because the printer works by extruding layers of plastic, thinner layers allow for more fine details.  However, more layers means that the part will take longer to print (all other things being equal).

Print Speed

3D printing is generally a slow process, but a faster print speed could definitely help offset that.

Nozzle Diameter

A small nozzle diameter means that the extruded plastic will be narrow and thus allow for fine details.  It also improves the smoothness of the printed object.

Ease of Removing Parts off the Platform

Because during a print, the base of the part (ideally) should be firmly adhered to the print bed, one must take care when removing the part from the print bed.  In some cases, removing a part requires a little bit of force to pry the part off the print bed.  This application of force may affect the alignment of the print bed relative to the nozzle.  In lower quality printers this can permanently affect the alignment or require frequent re-calibration   Seeing how “rigid” the print bed assembly can help assess how the machine accuracy can be affect during part removal.  For example, if the print bed is mounted on really large support structure, then the affect of misalignment is less of a problem than a machine where the print bed is mounted in a smaller support.

Enclosure

Some printers enclose the part to be printed in a box and some don’t.  The reason an enclosure exists is so that the machine is able to prevent the part being printed from losing too much heat during print.  Loss of heat during a print negatively affects the adhesion of the printed plastic.  Also a larger temperature difference means the part will warp more.  Without an enclosure, heat is able to freely transfer to the ambient.  High quality printers generally have an enclosure.

Variety of Printed Materials

A common material to print with is ABS.  Other materials such as PLA can be printed as well.  The use of the particular material can be quite dependent on the application of the printed part.  The cost of material also differs.  It is worthwhile to determine what material will be printed.  In many cases, ABS should be sufficient.

Warping

This is probably one of the biggest limitations of 3D printing.  Because of the use of heat during print, it is natural for the part to shrink as it cools down.  The problem is that the shrinkage is not uniform.  Some parts of the model shrink more than others and most of the time it is very unpredictable.  While there are some general guidelines that designers can follow to reduce warping, there are definitely some machine features that are good to have to help reduce warping even further.  Because the nature of the way the printer prints is that plastic is extruded layer by layer, the plastic layer at the very bottom is the first to cool down.  Cooling down means shrinkage.  This also means that as the bottom of the model shrinks, the printer is still printing on top of a slightly shrunken part.  Sometimes the shrinkage can be so bad that the part pulls itself off the print bed.  When the part is no longer securely adhered to the print bed, the part may slide around and ruin the entire print.  Warping is the enemy of 3D printing.

Below are some qualities of a machine that help reduce shrinkage:

• Heated print bed – a heated print bed keeps the plastic hot during the print and does not let the base cool down as much as if there were no heated print bed
• Enclosure – described above

There are more ways to reduce warping that the designer can do, but that is outside the scope of this article.

Number of Nozzles

A lot of printers just have the one print nozzle, but there are some with multiple.  This allows printing with multiple colours or in some cases printing with a water soluble support material.  Generally, support material has to be manually removed from the printed part, but if the support material is water soluble, this saves a lot of the manual labor involved in cleaning up the model.

Nozzle Maintenance

Through extended use, the nozzle can become quite dirty, that is black gunk starts building up on the surface of the nozzle.  If one is not careful, this can clog the print nozzle.  It is worthwhile to find out how often the nozzle needs to be replaced.  There are also other factors that may result in the need to replace a nozzle.  This is a recurring cost of maintaining the printer.  Some nozzles are cheaper and more common than others and it is worth it to see how often the nozzle needs to be replaced and how much it will cost each time.  Machines may come with extra nozzles.

Lubrication

Over time, moving parts may have to be lubricated.  Printers should come with a maintenance manual to help instruct the user how to do this.  It is also helpful to know the cost of the lubricant and how often it will be required.

Machine Lifetime Cost

Aside from the cost of the machine, operating a 3D printer can also incur the following costs:

• Shipping
• Insurance
• Brokerage fees
• Tools for removing support material
• Currency conversion
• Sales tax
• Print material (recurring)
• Replacement nozzles (recurring)
• Lubricants for moving parts (recurring)

As it can be seen, the extra costs can be quite significant.

Conclusion

Hopefully this has given a good general overview of some of the considerations in buying a 3D printer.  Different people will value different things in the printer and hopefully this subset of possible considerations for buying a 3D printer helps people make a decision as to what to look for in a printer.

Since I had no computer, I started randomly taking pictures and one of them are of the different coinage designs I had on me at the time…

In order as if reading normally:

1. One Euro: France
2. One Euro: Germany
3. One Euro: Italy
4. Two Euros: Italy
5. Dime: Spain
6. Dime: The Netherlands
7. 50c: Italy
8. Dime: Germany

• Introduction
• Motivation
• Newton’s Method
• Steepest Descent Method
• Limitation
• Sources

Introduction

In many sciences, the problem of non-linear optimization appears quite often.  What happens if they’re non-linear?  Sometimes, the system of equations remains easy to solve even given non-linearity, but other times, the possibility of solving the problem analytically may not exist if the algebra gets too complicated.  The solution lies in using numerical methods (approximations).  Here, I present two common methods: Newton’s Method, and the Steepest Descent Method. 

Each method has its own limitations and the choice of which method to use depends on the problem.  These methods involve using iterative computation in order to reach a solution.  Luckily, computers can prove useful in providing solutions very quickly due to its quick processing power for numerical computation.

In order to understand some of the concepts and terms used in this article, the reader has knowledge in multivariable calculus.  Also, equations in picture form made the line spacing look funny, so I’m going to do my best write the equations as text.  I should also point out that I couldn’t write the vector arrow on top of the variables in text form so vectors are written in bold.

Motivation

In calculus, the idea of optimization involves taking the derivative of the function and equating it to zero.  In multivariable calculus, the same idea applies except, we set all the partial derivatives to zero and solve the system.

So in general, if we are given a function:

ƒ(x), where x = [x₁, x₂, … x_n]^T

The maximum or minimum can be found by taking a partial derivative with respect to the vector x.  This gives the following equations:

∂ƒ/dx₁ = 0

∂ƒ/dx₂ = 0

…

∂ƒ/dx_n = 0

A short-hand way of expressing the above equations is by using the gradient vector, giving∇ƒ(x) = 0.  By definition, the gradient is defined by:

∇ƒ(x) = [∂ƒ/dx₁, ∂ƒ/dx₂, … , ∂ƒ/dx_n].

An analytical solution is not always possible, hence the motivation for approximating the solution using numerical methods.

Newton’s Method

In general, this method is written below.  Explanation of each step will follow.

1. Given the equations to solve, derive equations for the approximating line or plane, depending on whether the problem is 2D, 3D, nD, etc.
2. Using the linearized equations, solve the system of equations
3. Using the results from step 2, generate new equations for a new approximating line or plane, and repeat step 2.

This method also involves starting with a guess, denoted by a variable with a subscript of 0.  A guess is required, because this is the starting point for this method.  Using this guess, each successive solution should converge (more on that later).

Step 1

In order to solve the equations ∇ƒ(x) = 0 (see Motivation for more information), we can apply multivariate Newton’s method to approximate the solution.  Like the single-variate (two-dimensional) version, a linear approximation is required.  However, instead of having an approximation in the form of a line, the approximation is a plane instead.

2D (line): ƒ(x) ≈ ƒ(x₀) + ƒ’(x₀)(x – x₀)

3D (plane): ƒ(x, y) ≈ ƒ(x₀, y₀) + ƒ_x(x₀, y₀)(x – x₀) + ƒ_y(x₀, y₀)(y – y₀)

nD (nD plane): ƒ(x) ≈ ƒ(x₀) + ∇ƒ(x₀)(x – x₀)

Since we are trying to optimize, we have the partial derivatives of the function we are trying to optimize, but in order to apply Newton’s Method, we need a linear approximation of the partial derivatives.  This means that in order to approximate the partial derivatives, we need to take another derivative in order to get the slopes of the approximating planes.  If that description didn’t make sense, try the following.  For an equation with two independent variables, there will be two equations for the partial derivatives, one for x and one for y.  Taking partial derivatives from each of those equations again will give the slopes of the plane that approximate the first partial derivative.  So if we have two independent variables, we end up getting four equations: ƒ_xx, ƒ_xy,ƒ_yy, and ƒ_yx (ƒ_xy and ƒ_yx should be the same).  In general, for n partial derivatives, we would have n² equations if we took another set of derivatives.  The approximations for both of the partial derivatives would be:

ƒ_x(x, y) ≈ ƒ_x(x₀, y₀) + ƒ_xx(x₀, y₀)(x – x₀) + ƒ_xy(x₀, y₀)(y – y₀)

ƒ_y(x, y) ≈ ƒ_y(x₀, y₀) + ƒ_yy(x₀, y₀)(x – x₀) + ƒ_yx(x₀, y₀)(y – y₀)

Plug in the values for the second derivatives, and the values from the initial guess (the x y values and what function value you get using those values).

Step 2

Although the equations generated from step 1 is yet another system of equations, the main difference is that the equations from step 1 are linear.  This means that the system can be solved analytically.  Solve for the solution of the system using row reduction methods, or other methods.  The solution from solving the system will be the values used for the next iteration and will replace the guessed values.  The solution may or may not be close to the actual solution.  More iterations need to be done to see if the answer changes with each iteration.  Usually if the guess is really far away, then it takes a lot of iterations to get it really close.  However, sometimes, a bad guess may cause the solution to diverge, meaning that the solution will be further and further off.

Step 3

Generate new equations for the approximating planes using the solution from step 2.  Using these equations, go back to step 2 and solve it again.  This method is super tedious if done on paper.  Matlab can help, but then again, Matlab can do the entire approximation with just a few commands, but not everyone has access to it unfortunately.

Hopefully after a few iterations, the solution stops changing.  When that happens, you can substitute the final solution back into the original equation to see if it works!

Steepest Descent Method

Here is the general method:

1. Find the direction of the maximum or minimum directional derivative
2. Follow it until a maximum or minimum is reached

Newton’s Method is sufficient in a lot of cases, but sometimes it may be too hard to go beyond first derivatives.  This main idea of this method is to use the first derivative and follow the direction in which the function gets larger/smaller.  A fundamental concept used in this method is the directional derivative.  Since in 3D (or higher dimensions), we are not restricted to movement along the x axis.  This means we could move in any random direction, and in each direction the rate of change is going to be different.  To calculate the max or min, we need to calculate where the directional derivative is largest or smallest.  Just like a boulder rolling down the steepest slope on the hill.

Starting with a guess, we can find out the value of the function at the point and the direction to travel in.  Take a step in that direction and see how the value of the function changed.  Keep doing that until a max or min is reached.

Step 1

The definition of the directional derivative is the dot product of the gradient and the unit vector specifying the direction.  Here is the definition of the directional derivative:

D_uƒ = ∇ƒ(x)·u

Since the definition involves a dot product, it is true that if the two vectors are orthogonal, the resulting dot product is zero.  The direction that this derivative is in is called the level direction.  But if the two vectors are parallel, we get the highest value of the directional derivative.  For a vector that is parallel in the same direction, the directional derivative will be a maximum, but if the vector is antiparallel (parallel but both pointing opposite ways), the directional derivative will be a minimum.

Since the maximum value of the directional derivative is parallel to the gradient, we can calculate the unit vector of the gradient and use it as the unit vector u.  In order to get the minimum, simply flip the direction of the vector by multiplying it by –1.

u = ∇ƒ(x)/|∇ƒ(x)|

Although not necessarily required, substituting this value into the definition of the directional derivative, and doing some simplification with algebra, the maximum and minimum directional derivative is given by:

Max D_uƒ = |∇ƒ(x)|

Min D_uƒ = -|∇ƒ(x)|

Using the initial guess, calculate the direction of to travel in at that point.

Step 2

Knowing the direction to travel choose a desired step size “h”.  This means that if the equation will be traversed in a particular direction, we need to know exactly how far to go.  Choosing smaller step sizes gives better computational accuracy but increases computation time.

To get to the maximum, use the direction for the maximum directional derivative.  To get the minimum, use the direction for the smallest directional derivative.

Starting with a guess, calculate the function value at that point.  Then take a step in the direction given by Step 1 using the step size.  The variable “u” here is a component of the unit vector

x_n+1 = x_n + u_xh

y_n+1 = y_n + u_yh

At this new value, calculate the function value again.  Keep repeating this until the maximum or minimum is reached.

Limitation

The approximation is generally only as good as the initial guess.  Sometimes, a bad guess can lead to a solution that never converges.  In that case, a different value must be used in order to find a converging solution.  For example, if we have a fourth order polynomial function, a local extreme will exist.  As x approaches –∞ or ∞, y will do the same.  If I use the steepest descent and give a bad guess, my solution will go to ∞ instead of the local extreme.

Sources

1. Calculus by Gilbert Strang
2. Applied Regression Analysis by Norman Draper and Harry Smith

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Update 24 Sept 2011: Added info about screw head dimensions
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