Gaussian elimination is a method for solving matrix equations of the form

Future Perfect Tense	Future Perfect Continuous Tense
Form
& The affirmative form of the Future Perfect tense is formed by means of the auxiliary verb to have in the Future Indefinite and the Past Participle (Participle II) of the main verb. S + will have VIII/-ed…     e.g. I'll have read the story by the time you come back.   & The negative form of the Future Perfect tense is formed by means of the auxiliary verb to have in the Future Indefinite, the negative particle not and the Past Participle (Participle II) of the main verb.     S + will not have VIII/-ed… e.g. You won’t have forgotten me by then.   & General question in the Future Perfect tense is formed by means of the auxiliary verb to have in the Future Indefinite placed before the subject and the Past Participle (Participle II) of the main verb.   Will (Won’t)+ S + have VIII/-ed…? e.g. Will she have finished her work by 8 o’clock? Won’t she have finished her work by 8 o’clock? & Special question in the Future Perfect tense is formed by means of the special question word on the first place, the auxiliary verb to have in the Future Indefinite placed before the subject and the Past Participle (Participle II) of the main verb.   ? will (won’t)+ S + have VIII/-ed…? W e.g. By what time will (won’t) the meeting have finished? Who will (won’t) have taught you to speak French by the end of this month?	& The affirmative form of the Future Perfect Continuous tense is formed by means of the auxiliary verb to be in the Future Perfect and the Present Participle (Participle I) of the main verb. S + will have been V-ing …     e.g. I'll have been reading the story for two hours .by that time   & The negative form of the Future Perfect Continuous tense is formed by means of the auxiliary verb to be in the Future Perfect, the negative particle not and the Present Participle (Participle I) of the main verb.   S + will not have been V-ing … e.g. You won’t have been doing your homework for an hour by then.   & General question in the Future Perfect Continuous tense is formed by means of the auxiliary verb to be in the Future Perfect placed before the subject and the Present Participle (Participle I) of the main verb.   Will (Won’t)+ S + have been V-ing …?   e.g. Will she have been working here for tree years by the end of this year? Won’t she have been working here for tree years by the end of this year? & Special question in the Future Perfect Continuous tense is formed by means of the special question word on the first place, the auxiliary verb to be in the Future Perfect placed before the subject and the Present Participle (Participle I) of the main verb.   ? will (won’t)+ S + have been V-ing …? W e.g. How long will Simona have been staying with you by next Sunday? What will (won’t) Jason have been doing for five weeks by the end of the year?
Use
Ø The Future Perfect is used for an action which will be finished before a stated future time. e.g. They will have finished their meeting by four o'clock this afternoon.	Ø The Future Perfect Continuous is used to emphasise the duration of an action up to a certain time in the future. e.g. By the time Rick retires, he will have been working for the same company for thirty years. Ø The Future Perfect Continuous is not used with the state verbs. The Future Perfect is used instead. e.g. Fred will have owned his new car for ten weeks by the end of the year.
Note: After the time expressions by the time, until, before, we use the Present Simple because they introduce time clauses. The Future Perfect and The Future Perfect Continuous may come either before or after the time clause. Study the following examples: a) I will have tidied up by the time you get back. (time clause) b) By the time she finishes work, we will have been waiting for more than an hour. (time clause)
Time Expressions
The Future Perfect is used in sentences with the following time expressions: by, by the time, before, until, by then, etc.	The Future Perfect Continuous is used with: by ... for

EXERCISES

Exercise 1. A) The people below are all friends. What do they hope they will have done by the time they are thirty years old? Using the prompts below, make sentences, as in the example.

e.g. Alec hopes he will have travelled all over the world by the time he's thirty.

B) The same people will be thirty-five years old in a month's time. What will they have been doing by the end of next month? Using the prompts below, make sentences, as in the example

1 Alec / travel / ten years

...By the end of next month, Alec will have been travelling for ten years

2 Mark / run / own business / eight years

3 Moira / star in films / six years

4 Kate / design clothes / four years

5 Jack / build houses / ten years

Ted / paint / fifteen years

Exercise 2. Disagree as in the example.

e.g. Monica will have flown to Brazil by the dinner. – Oh, no. Monica won’t have flown to Brazil by the dinner.

1. Tom's parents will have dug the plot by the evening.

2. We will have signed the documents by two o'clock.

3. They will have crossed two oceans by September.

4. The tourists will have reached the camp by 6 p.m.

5. The ministers will have met by the afternoon.

6. Nick will have shot the film by the summer.

7. Hanna will have sewn her dress by the end of the week.

8. I will have baked this pie by 3 p.m.

1. Norma will have knitted the sweater by June.

Exercise 3. Ask as in the example.

e.g. Polly will have washed the cutlery by the dinner. – By what time will Polly have washed the cutlery?

1. The guests will have left by 11 p.m.

2. I will have cleaned the saucepans by the evening.

3. They will have watched the film by 10 p.m.

4. Billy will have caught some fish by breakfast.

5. Ada will have eaten all the grapes by my coming.

6. We will have taken all exams by the end of June.

7. The operation will have been over by 2 p.m.

8. I will have read all the press by that time.

1. Mom and I will have roasted the turkey by dinner.

Exercise 4. Ask and answer as in the example.

e.g. We will have been flying for eight hours before we get to Sidney. – Will you have been really flying for eight hours before you get to Sidney? – Oh, no. They won’t have been really flying for eight hours before they get to Sidney?

1. Next year Sheila will have been working for this firm for ten years.

2. In two months Barry will have been dancing at this theatre for five years.

3. In twenty minutes the students will have been writing the test for three hours.

4. Tomorrow it will have been snowing for three days.

5. Next summer the aircraft designers will have been designing this plane for three years.

6. Next week William will have been playing polo for this team for ten years.

7. In a few days Jessica will have been writing this book for a year.

Exercise 5. Put the sentences into Future Perfect Continuous and mind the time expressions.

1. I work for this firm twenty years. - By the tenth of April 1 shall have been working for this firm for twenty years.'

2. It often snows in December.

3. Helen plays the piano every week.

4. We redecorate the rooms every five years.

5. The mechanic fixes the car.

6. They repair the water pipes.

7. Sean looks for a job.

8. Rachel paints the portraits.

9. Alan and I play hockey for this team.

10. My relatives solve this problem.

11. Our aunts weave the carpets all their life.

12. Mark's dad teaches French for many years.

13. Joe has been squeezing the lemon for ten minutes.

14. Jill has rented this flat for five years.

15. Neil stews the meat every day.

Exercise 6. Put the verbs in brackets into the Future Perfect or the Future Perfect Continuous.

1. By 3 o'clock, she ...will have been studying... (study) for six hours.

2. By the end of next month, Sam … (finish) the project.

3. He … (not/start) painting the kitchen before Tuesday.

4. By the time she arrives in Paris, she … (travel) for four hours.

5. I hope I … (buy) my own house by the time I'm thirty-five.

6. By Saturday, Lisa … (diet) for two weeks.

7. Hopefully, they … (learn) everything by the time they sit the exam.

8. By 4 o'clock, I … (sit) in the hairdresser's for three hours.

9. By Christmas, I … (work) for this company for eighteen months.

10. By next weekend, Brian … (move) house.

11. Hopefully, the builders … (finish) building the house by next month.

12. By Tuesday, Alan … (sail) for twelve days.

13. By tomorrow morning, she …. (sleep) for twelve hours.

14. By 7.00 pm they ... will have been playing… (play) cricket for eight hours.

15. I … (finish) painting your room by the time you get home.

16. By the end of next month I … (live) in London for exactly three years.

17. Tom ... (write) his third novel by the end of this year.

18. By the time he arrives in London. John … (drive) for five hours.

19. This film … (probably/not/finish) until midnight.

20. How long … (you/study) English by the end of this term?

21. Don't worry! You … (forget) all about it by this time next year.

22. By Friday I … (work) on this project for two weeks.

23. Hopefully, he ... (cook) dinner for us by the time we get home.

24. I … (read) this book by tomorrow night.

25. He hopes he ... (make) a million pounds by the time he is thirty.

26. By 5 o'clock I … (do) this crossword puzzle for three hours.

Exercise 7. Complete the sentence with either the Future Perfect or the Future Perfect Continuous.

e.g. Simon started to learn Spanish when he was 25. He is still learning Spanish. – When he's 40 he will have been learning Spanish for 15 years.

1. Every day, Peter eats three bars of chocolate on the way home from school. – Before he gets home from school tonight, Peter …

2. So many people enter the New York Marathon that the last runners start several minutes after the ones at the front. – By the time the last runners start the ones at the front ...

3. I started writing this book three years ago next month. – By next month, I ....

4. The company is spending $5 million on developing the software before it goes on sale. – By the time the software goes on sale, the company...

5. I’m going to paint the front door today. I'll finish it before you get back. – When you get back, I …

Exercise 8. Tell your group mates about:

a) 10 things that you will have done in 5 years' time

b) 10 things that you will have been doing for the next 5 years' time by the time you graduate the University.

HOME TASK

Exercise 9. Put the verbs in brackets into the Future Perfect Continuous or the Future Perfect.

1. A: I can't come shopping on Saturday morning because I ...will/'ll be working... (work).

B: That's a pity.

2. A: Don't phone me later than midnight because I … (sleep) then.

B: Shall I give you a call at about 10:30, then?

3. 3 A: Come to my house at six o'clock.

B: … (you/finish) you homework by then?

4. A: Have you made the preparations for the party?

..... B: Not yet, but I … (finish) them by this evening.

5. A: … (you/go) to James' party on Saturday night?

..... B: Yes. A: Would you mind giving me a lift?

6. A: There's a meeting tomorrow at 4 o'clock.

B: I can't go if it is that late. I … (leave) by then.

7. I … (cook) dinner by the time you get home.

8. In a few minutes, I … (wait) here for Craig for over two hours. Where can he be?

9. Weil be halfway through the sponsored swim in one hour, so we … (swim) for forty-eight hours non-stop by then.

10. They … (not / finish) painting the house by the time we get back from holiday.

11. If she's still on the phone at eight o'clock, … (Jan / talk) to Melissa for over two hours.

12. At six o'clock this evening, we … (not / climb) for five hours but for seven hours!

13. This time next month, you … (probably / pass) your driving test!

14. …. (you / do) all your homework by bedtime?

15. Tomorrow … (Elaine / work) on the project for ten days.

16. We …. (probably / not / leave) by the time you get home.

Exercise 10. Put the verbs in brackets either into the Future Continuous or Future Perfect Continuous tense.

1.

Jack: Have you been watching the Eco-Challenge on TV?

Janet: Isn't that exciting? It has got to be the most unbelievably difficult sporting event in the world.

Jack: I know. By the time they finish the course, they (raft) more than 150 miles down a raging river, (hike) through 80 miles of jungle, (climb) a volcano and (kayak) through shark-infested waters.

Janet: And don't forget that they (move) for at least eight days straight.

2.

Oliver: When are going to get your bachelor's degree, Anne?

Anne: I am going to finish my degree next June. By the time I graduate, I (go) to four different colleges and universities, and I (study) for more than seven years.

Oliver: Wow, that's a long time!

Anne: And I plan to continue on to get a Ph.D.

Oliver: Really? How long is that going to take?

Anne: By the time I finally finish studying, I (be) a student for over 13 years.

3.

Max: Sarah has been in the kitchen all day long.

Jake: It doesn't sound like she's having a very good Thanksgiving.

Max: She (cook) for over seven hours by the time everyone arrives for dinner this afternoon. Hopefully, she (finish) everything by then.

Jake: Maybe we should help her out.

4.

Fred: By the time they finish their trip across Yosemite National Park, they (hike) for more than six days.

Ginger: And they (be, not) in a bed or (have) a shower in almost a week!

Fred: When we pick them up, they (eat) camping food for days, and I am sure they will be starving.

Ginger: I think we had better plan on taking them directly to a restaurant.

Exercise 12. Write 15-20 things of changes in the world you expect by the end of the 22d century. Don’t forget to use either the Future Perfect or Future Perfect Continuous tense in each sentence.

Gaussian elimination is a method for solving matrix equations of the form

(1)

To perform Gaussian elimination starting with the system of equations

(2)

compose the "augmented matrix equation"

(3)

Here, the column vector in the variables is carried along for labeling the matrix rows. Now, perform elementary row operations to put the augmented matrix into the upper triangular form

(4)

Solve the equation of the th row for , then substitute back into the equation of the st row to obtain a solution for , etc., according to the formula

(5)

In Mathematica, RowReduce

LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.

A matrix that has undergone Gaussian elimination is said to be in echelon form.

For example, consider the matrix equation

(6)

In augmented form, this becomes

(7)

Switching the first and third rows (without switching the elements in the right-hand column vector) gives

(8)

Subtracting 9 times the first row from the third row gives

(9)

Subtracting 4 times the first row from the second row gives

(10)

Finally, adding times the second row to the third row gives

(11)

Restoring the transformed matrix equation gives

(12)

which can be solved immediately to give , back-substituting to obtain (which actually follows trivially in this example), and then again back-substituting to find

Date: 2015-01-02; view: 3439