How can I get rid of this ivy cheaply and quickly (picture below)?

snip the root!!

Not familiar with this variety of ivy. Dig it out making sure to get all the roots. It is still small, so it shouldn't be to difficult.

Keep it cut back to the ground and remove the rest of the ivy, it will get under your siding. Hopefully with time and diligence of keeping it cut back the lack of sunlight will kill the roots.

That is not ivy. It is trumpet vine. It is very invasive. The root has to be dug out to get rid of it. Any part of the root left will come up again

It is Virginia Creeper. The roots run deep and you will find out you will have more next year. When you weed what you can the roots underground send out runners. Weed the best you can and spray where it goes into the ground with ground clear. Just pull it off of the house now. It will take vigilance to weed any new shoots coming out of the ground.

I wanted to get rid of a tree stump because it kept sprouting little shoots. So heard about epson sales as a killer. I drilled holes in the top and wet it good then put epson salt in the holes. I did this every 3 days for a couple weeks. Killed dead. So I'm suggesting you try this. Cut them down so the branches are exposed and mix a strong solution of water and Epsomn salt . Saturate the area every few days and put dry epson on them too. See what happens. No poison to worry about.

It looks like Virginis Creeper or Creeping Ginny. Either is highly invasive and hard to kill. I've tried everything from RoundUp to boiling water. Neither were effective. I've even tried spraying it with jalapeno pepper water. You have to dig out the underground roots! I suggest you cut it a few inches up from the ground (so you can still tell where the roots are), pull the climbing/running foliage down, then dig out the roots. You can put table salt in the hole IF there is nothing growing nearby that you want to keep. It is important to say that if you do use the salt, nothing will grow there for a very long period of time. A substitute for salt is Boric Acid which also 'kills the soil' so nothing can grow in it. Both are pretty drastic measures but sometimes necessary because it will damage your siding or bricks. Important! Creepers have 5 leaves, poison ivy/oak has 3 leaves per cluster and they look a lot alike so be vigilant!

I cut mine down to the ground, raked, the soil, making sure I left some holes in the ground, sprayed with natural weed killer. , after a few days covered the area with black landscape fabric, and mulch. Hasn't grown back. Requires light and sun to grow.

Additional comment: someone suggested using Epsom Salt, which I agree will work. Just be sure you use a very generous amount. If you use just a sprinkle, it will act as a fertilizer!

Wow, lots of answers and some are great. The vine in the picture is not ivy, trumpet vine or creeping jenny, it's called Virginia creeper and unfortunately, it's terribly invasive. I have the vile stuff growing on one side of my yard on a fence. Rip it all the way to where it goes into the ground. I'm not at all a huge proponent of using Round Up, but in this case, I think that's the only thing that will get rid of it. Inside of spraying, try painting just the leaves and stem at the root. Stay on it and hopefully you can eradicate it. Good luck!