I call my method - a Petrus first two layers (F2L) and Fridrich last layer (LL) - the Petrich method. In this method, you orient edges (flip bad edges) before you reach the LL during the Petrus method step 3. When I reach the LL, instead of positioning (permuting) corners, I just orient all the cubies the same way so that the top is a solid color (like in Allan, H-permutation, Z-permutation, A-permutation (three-corner cycle), and Niklas/Sune (J-permutation) positions). Then, once the top is a solid color, I just position (permute) all the pieces in one algorithm.

I believe using this approach for the LL is faster because you save a lot of recognition time. When I reach the LL, I do not have to worry about how corners and edges have cycled around, I simply look at the pattern (whether it's a Sune, Bruno, Double Sune, Triple Sune, or two Sune position (two correct corners on heels and two correct corners on toes)) and apply one of 7 algorithms (Sune, mirrored Sune, Bruno, Double Sune, and three new algorithms that are only 8, 8, and 10 turns long for solving the Triple Sune, two correct corners on heels, and two correct corners on toes positions, respectively), 4 of which you already know, to orient all the cubies the same way so that the top is a solid color. After you have made the top a solid color, only then will you need to spend a bit more time recognizing the position. However, it is still faster, in my opinion, than recognizing which corners have swapped and which edges have cycled. This is because since the top is already a solid color, you can just ignore the top and look at the sides. When the top is not a solid color, it is harder to see where all the colors are and identify which corners and edges have cycled.

I have calculated (or rather just plain counted) the total number of algorithms you need to memorize for various two-look (two-step) LL methods when you use a Petrus F2L and the results clearly show that my Petrich method requires the least possible memorization for a two-look LL with edges already oriented (all good edges). Here are a few examples, counting all mirrored algorithms and inverse algorithms:

The first two-look LL method with a Petrus F2L I learned was the Petrus two-look (5, 6+7). Then I learned the Fridrich two-look LL method with a Petrus F2L. I also learned the Fridrich two-look LL with a Fridrich F2L, which required me to memorize 57 (OLL) + 21 (PLL) = 78 algorithms, most of the algorithms of which I have already forgotten because in order for a full Fridrich method (Fridrich F2L + Fridrich LL (d'oh)) to be effective, one must also learn numerous F2L algorithms and get used to building a cross, the latter two of which I have not accomplished. Perhaps when I have time, I will go learn all those F2L algorithms and practice building a cross, and then I will switch to a full Fridrich method. For now, I believe my Petrich method is the most practical approach for anyone who wants to become a speedcubist and also has a (busy) life. Clearly as you can see from the memorization comparison along with my recognition paragraph above, the Fridrich LL goes very well with a Petrus F2L and it is possible to achieve very good times with this method (as my times have demonstrated).

Now that I have given you an overview of what my Petrich method looks like, I will go on to describe several things I did to further improve my Petrich method:

1. Experiment with several algorithms for each case and choose the best one. ***Shorter does not necessarily mean faster.*** If you are unsure what algorithms to use, I suggest to visit the websites of all the top cubers in the world and learn one of the algorithms they use. These top cubers have already gone through all the hard work of experimenting with several algorithms for each case and have already picked out a handful of great algorithms for each case for you. However, just because one of the best cubers in the world uses a particular algorithm does not necessarily mean that that algorithm will suit you. Your hands may be different, and the types of triggers you like may be different. The algorithms may involve double-layer turns or cube rotations that you may not like. Also, you may not understand some of the fingertrick notation used, such as parentheses and grip change notation. If you want an explanation of this notation, you will probably be able to find an explanation on the cuber's site, but if not, you can always ask me :-).
2. Learn the "step 4b tricks" at . Many of the cases are intuitive, but some are more complex and require memorization of an algorithm. I have memorized all of them and currently actively use all of them except 3 (including mirrors), which I need to practice more before I will feel comfortable enough using them in speedsolves. Make sure not to forget to practice mirrored algorithms.
3. Learn some of the optimal (pure, least number of turns) solutions for solving 2, 4, and 6 bad edges in step 3. 2 bad edges should take a maximum of 5 turns to solve, 4 bad edges should take a maximum 6 turns, and 6 bad edges 8. I did not memorize all of these optimal solutions, but glanced over some of them so that I am now able to intuitively perform an optimal solution for step 3 the majority of the time. Take a little bit of time to study this step and you will surely find ways to minimize turns. All optimal solutions are very simple and straightforward except the 6 bad edges positions (of which there are only 3, of which all use the same basic 7-turn move with 2 of the 3 positions having an extra turn added in front of the 7-turn move). Once you "understand" how the optimal solutions work, you should be able to generate them intuitively.
4. Practice step 4 (a and b) and its algorithms from different angles to reduce grip changes, which cost you time. We all speedsolve "right-handed" so when we get to step 4, we are holding the 2x2x3 in the bottom left with our left hand and can do the R and U turns freely with our right hand and left hand index finger. We all approach step 4 by first building a 1x2x2 block (4a) and then adding in the last two pieces (4b), generally as a pair (together, at the same time, not one at a time). Thus, there are 4 different possible 1x2x2s we can build and 4 locations for step 4b. When doing the last few turns of step 3, you want to be already looking ahead for the easiest 1x2x2 to build in step 4a. You need to practice building these 1x2x2s both on the top (U) side and the right (R) side and front and back and practice executing step 4b algorithms at these locations. This is something I need to work on myself and I think I will practice this more now that I have written it down and realized I do not practice this enough.
5. Search for shorter, more optimal solutions for the 2x2x2 and 2x2x3. You should always be able see every turn needed to build the 2x2x2 before you start during the 15 seconds inspection. My approach is to start off looking for 1x1x2 blocks, which generally determine the 2x2x2 I will be building. Sometimes, when I find several 1x1x2 blocks or even a 1x2x2 block, I will directly build a 2x2x3. A direct 2x2x3 build usually involves me building a 1x2x3 block, then solving the two remaining edge pieces, and finally moving the 1x2x3 block in place. Choosing to build a 2x2x2 where the corner piece is already in the correct position and oriented (twisted) correctly is generally not a good idea unless it is also part of a 1x1x2 or greater block. This is because you cannot solve any of the other 3 edges without breaking apart the corner. Thus, you usually end up using a 3-turn move of moving the corner piece away (thus protecting it), moving an edge piece in place, and then moving the corner back in place for each of the 3 edges, adding up to 9 turns. This is not even counting the additional turns that may be necessary in order to line up one of the facelets on an edge piece with its corresponding center piece. Thus, this approach will most likely produce a 2x2x2 requiring 10+ turns. When doing thinking/fewest moves (FM) solves (I do not time myself; I simply search for the optimal solution), I average around 7 turns for the 2x2x2. With practice, you should almost always be able to see a 2x2x2 solution that is less than 10 turns during your 15 seconds inspection. Optimizing the 2x2x3 build, or rather the 1x2x2 build that you add on to the 2x2x2 is something I will save for future discussions after I have spent more time analyzing this step. Step 2, the 2x2x3, is probably the most difficult step to practice and improve, and I will explain in the future.
6. Learn and practice new triggers and try to add these triggers into your algorithm executions. Get more fingers involved. Since we are all "right-handed" cubers, two fingers I believe we do not use enough are the left hand index finger and the right hand ring finger. The left hand index finger comes in handy a lot such as in step 4 and also in many triggers, such as my repetitive Triple Sune trigger and the 2nd movement of the Z-permutation. The left hand index finger can also be used in executing a half turn. "How?" you may ask. Consider the common trigger R, U2. One way of executing this trigger would be to start with your thumb on the D face and your index and middle fingers on the U face and making the R turn with a turn of your wrist and then making the U2 turn with an index-to-middle finger half turn. However, consider this alternative to executing the R, U2 trigger. Start the same way with your thumb on the D face and your index and middle fingers on the U face. Make the R turn the same way by turning your wrist. Then, keep pushing with your thumb in order to do a U' turn. Immediately after the U' turn finishes, make another U' turn with your left hand index finger. With practice, the U2 turn will not seem like two separate U' turns, but rather a fluid U2 turn. You may ask how in the world the right hand ring finger can be used. Consider Alan Jiang's two-movement Sune. He does the first four turns of the Sune (R, U, R', U) in one movement by doing the common R, U, R' trigger and then dragging the U turn with his ring finger. Thus, the second U turn becomes part of the R, U, R' trigger and you do not lose time shifting your grip. Several Fridrich algorithms use the (R, U, R', U) trigger or variations of it and the ring finger can be used in all of them. Another example I can give where the ring finger may be useful is very early on when you are building your 2x2x2. Say you needed to do U', D' to shift two or more pieces to where you wanted. While your left hand index finger is doing the U' turn, your right hand ring finger can be doing the D' turn. The ring fingers are also heavily used in one handed cubing. This I can explain in the future or in another e-mail if you like.
7. Do thinking/FM solves. Don't just keep taking averages. Your improvement, if any, will be slow that way. In order to improve, you need to spend your cubing time wisely and practice things you have difficulty with or need to improve on. When doing thinking/FM solves, don't time yourself. Take all the time in the world, but always be thinking. Feel free to experiment. Thinking/FM solves are where you get to try out new things apart from your orthodox speedsolving method. You'll discover new things and find shorter solutions to positions which you thought couldn't be shortened. When I asked the Petrus Method Group for suggestions on getting sub-20, Lars Petrus personally suggested doing thinking/FM solves. He told me that when he was young, he would spend more than half his time doing thinking/FM solves and that was what got him so fast. I took his advice and did a lot more thinking/FM practice than I had been doing, and what do you know? In less than a week, I achieved my dream goal of a sub-20 second average. This really works.

This concludes my tips for improvement section. Wow, I am very happy to have written all this down. This is what I have meant to tell all my speedcubing friends, but I have never totally had the chance. Now I have done it. That was very satisfying =).

Something I learned the hard way, mainly because I lack the common sense, was to NEVER ever use your best speedcube for underwater cubing. Bad idea. A big no-no. Ruining your best speedcube is not the way to celebrate your first sub-20 second average, as I did. Okay, so from now on, I'll always be using a crappy cube for underwater cubing. I did do an EXTENSIVE, massive, and meticulous clean of my best speedcube and then lubricated it with silicone spray, and it is much better, though I do not think it is as good as it was before the underwater cubing. I did, however, set a new fastest one-handed (right) solve of 40.64 and a new fastest one-handed (left) average of 58.75 (sub-1 minute!). However, I think my 3x3 speedsolving has deteriorated with my lack of practice due to incompetent cubes. My 4x4 speedsolving has improved however, with a new fastest solve of 1:39.52 (sub-2 minutes!) and a new fastest average of 2:11.61. I can still easily average sub-25 seconds on 3x3 speedsolving, though, so I ain't too bad.