1、数值分析第五版李庆扬王能超易大义主编课后习题答案数值分析第五版_李庆扬_王能超_易大义主编课后习题答案（The fifth edition of numerical analysis _ li qingyang _ yi dayi is the chief editor）Chapter I introductionThe relative error of the set is the error of finding.Solution: the relative error of approximate values isAnd the error isAnd then there are2
2、. The relative error of the set is 2%, the relative error of the request.Solution: set, the functions condition number isAgain,againAnd for the 23. The following are the approximate Numbers that have been rounded and rounded, i.e. the error limit is not more than half of the last one, and the test i
3、ndicates that they are several valid digits:,Solution: five valid digits;Its two valid digits;Four valid digits;Five valid digits;Its two valid digits.4. Use formula (2.3) to find the following error limits (1), (2), (3).These are the Numbers given in question 3.Solution:5. When calculating the volu
4、me of the ball, the relative error limit is 1. What is the relative error limit allowed when measuring the radius R?Solution: sphere volume isThe condition number of the function isagainTherefore, the relative error limit allowed when radius R is measured6. Set, press the formula (n = 1, 2,.)To the
5、calculation. If you take (5 valid digits), how much error do you have in the test?Solution:.And then you plug in, you haveThat is,If take,The error limit is.7. Find two roots of the equation and make it have at least four valid digits ().Solution:,So the root of the equation should beTherefore,It ha
6、s five valid digitsIt has five valid digits8. What do you do when the N is sufficiently large?solutionSet.the9. The length of the square is about 100cm, so how can the area error be not exceeded?Solution: the square area function is.When, if,theTherefore, the area error can not exceed 0.005 cmLets s
7、ay that the g is accurate, and the measurement of t is a second error, proving that the absolute error of S increases when t increases, but the relative error decreases.Solution:The absolute error increases when increasingWhen the increase is maintained, the relative error is reduced.11. The sequenc
8、e satisfies the recursive relationship (n = 1, 2,.). ,If (three valid digits), how large is the error? Is this computational process stable?Solution:againagainThe calculation of the time error is that the calculation process is unstable.12. Calculate, take, use the following equation to calculate, w
9、hich get the best result?,.Solution: set,If so, then.If you compute the y value, thenIf you compute the y value, thenIf you compute the y value, thenThe result is best calculated.13. The value of the request. So if we square it with 6 tables, what is the error in the logarithmic time? If you switch
10、to another equivalent formula.What is the error in the log?solution,settheTherefore,If you use the equivalent formulatheAt this point,Chapter two interpolation method1. Quadratic interpolation polynomial in time.Solution:The quadratic Lagrangian interpolation polynomial is2. Numerical tableX 0.4 0.5
11、 0.6 0.7 0.8LNX - 0.916291-0.693147-0.510826-0.356675-0.223144Approximation of linear interpolation and quadratic interpolation.Solution: by form,If you use linear interpolation,theIf the quadratic interpolation method is used,3. To complete the function table, the step length function table has fiv
12、e valid digits, and the total error bounds when the linear interpolation is used to find the approximate value.Solution: when solving approximate values, the error can be divided into two parts. On the one hand, x is approximate and has five valid digits, resulting in a certain error propagation in
13、the subsequent calculation. On the other hand, using interpolation method to find the approximate value of the function, the interpolation method with linear interpolation method is not 0, and there will be some error. Therefore, the calculation of total error bounds should be combined with the abov
14、e two factors.When,maketakemaketheWhen, linear interpolation polynomial isThe interpolation of the remainder isWhen setting up the function table, the data in the table has five valid digits, and the error propagation process is in the calculation.Total error boundsSet to the different nodes, please
15、:(1)(2)prove(1) theIf the interpolation node is, the subinterpolation polynomial of the function is.The interpolation of the remainder isagainWe can see from the above conclusionHave to pass.5.Solution: order is the interpolation node, and the linear interpolation polynomial is=The interpolation is
16、zero6. In the table of equidistant node functions given, if the approximation of quadratic interpolation is used, the truncation error should not be exceeded, and how much should the step length of the function table be used?Solution: if the interpolation node is and, the interpolation of the quadra
17、tic interpolation polynomial isThe step length is h, that isIf the truncation error does not exceed, then7. If,Solution: according to the definition of the forward difference operator and the central difference operator.8. If it is m degree polynomial, remember that the k - order difference is polyn
18、omial, and (positive integer).Solution: the display of the function isAmong themIts a polynomial of the number of timesPolynomial of orderPolynomial of orderIn this process, its a polynomialIs constantWhen its a positive integer,9. To proveproveHave to the10. ProveProof: it is known from the above c
19、onclusionHave to pass.11. To proveproveHave to pass.12. If there is a different root,Proof:Proof: there is a different real rootandmakethewhilemaketheagainHave to pass.13. The following properties of the proof order are:(1) if(2) if, thenProof:(1)Have to pass.+Have to pass.14. Please.Solution:ifthe1
20、5. It is shown that the remainder of the Hermite interpolation of two points isSolution:If, the interpolation polynomial satisfies the conditionThe interpolation is zeroThe interpolation condition can be seenandCan be written asIts about the undetermined function,Now lets think of a fixed point as a
21、 functionAccording to the properties of the remainder, yesThe rolles theorem tells us that there are and thatThat is, there are four different zeros.According to rolles theorem, at least one zero point between the two zeros,So there are at least three different zeros in the inside,With this kind of
22、push, there is at least one zero inside.Remember to keep theagainDepends onIf the node is set at three times, the step length isIn the neighborhoodFind a polynomial P (x) that is no more than 4 times to satisfy itSolution: use the emir interpolation to obtain a polynomial of no higher than 4setWhere
23、, A is the undetermined constantthus17. Set up and take a piecewise linear interpolation function at the isometric node to calculate the value and error of the mid-point of each node.Solution:ifThe step lengthIn the interplot, the piecewise linear interpolation function isThe value at the middle poi
24、nt of each node isWhen,When,When,When,When,erroragainmakeThe stagnation point is the sum18. Find the linear interpolation function in the upper segment and estimate the error.Solution:On the interval,The function is a linear interpolation function in the intersectionsError is19. Please interpolate t
25、he emmett in the upper section and estimate the error.Solution:On the interval,makeThe function is a segment Hermite interpolation function in the intervalError isagain20. The given data table is as follows:Xj 0.30 0.39 0.45 0.53Yj 0.5477 0.6245 0.6708 0.7280Try three spline interpolation and meet t
26、he conditions:Solution:The matrix form of the system is2 1 M02 M12 M22 M31 2 M4Solve this systemThe cubic spline expression isTo plug inThe system of the matrix starts from this matrix isSolve this systemAnd then we have three more spline expressionsTo plug in21. If it is a cubic spline function, th
27、e proof:If, in the form of interpolation node, and, thenProof:Thus there areThe function approximation and the curve match1.,Give the Bernstein polynomials.Solution:Bernstein polynomial isAmong themWhen,When,When you are, ask for evidenceProof:If,The proof function is linearly independentProof:ifTak
28、e the inner product of the upper and the upper ends, respectivelyThe coefficient matrix of this system is Hilbert matrix, and the symmetry is positive and nonsingular,So we have zero solution a is equal to 0.The function is linearly independent.4. Calculate the following functions:M and n are positi
29、ve integers,Solution:If,Its monotonically increasingIf,If m and n are positive integersWhen,When,Its decreasing in the inner monotoneWhen,Its decreasing in the inner monotone.ifWhen,Its decreasing in the inner monotone.5. proveProof:6. To define,Ask them if they form inner product.Solution:(C is con
30、stant, and)thewhileThis is contradictory when and only whenCannot constitute the inner product.If,If,And,That is, when and only when.So you can form the inner product.7. So, the test is the orthogonal polynomials of the upper right.Solution:If,Order, then, and thereforeAnd then chebyshev polynomials
31、 are orthogonal to each other on the interval, andIts an orthogonal polynomial with right.again8. For the right function, the interval, try to find the orthogonal polynomial of 1Solution:If, the interval is the inner productDefinition,Among them9. It is proved that the second type of chebyshev polynomials given by the textbook formula are orthogonal polynomials of upper belt
