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Chapter 7: Problem 1

Find the three-digit decimal floating-point representation of each of the following numbers: (a) 2312 (b) 32.56 (c) 0.01277 (d) 82,431

Short Answer

Expert verified
The three-digit decimal floating-point representations for the given numbers are: (a) 2312.000 (b) 32.560 (c) 0.013 (d) 82431.000

Step by step solution

01

a) 2312

Since 2312 is a whole number and has no decimal places, we can simply represent it as 2312.000. After rounding it to three decimal places, the floating point representation stays the same, which is 2312.000.
02

b) 32.56

The number 32.56 already has two decimal places. In order to represent it with three decimal places, we only need to add one more decimal place. Since there are no other digits after the "6", we can simply add a "0" at the end, so the three-digit decimal floating-point representation is 32.560.
03

c) 0.01277

For the number 0.01277, we need to round it to three decimal places. To round a number, we look at the digit immediately after the third decimal place, which is the number "7" in this case. Since "7" is greater than or equal to 5, we'll round up the previous digit ("7") by 1, and the rounded number will become 0.013. The three-digit decimal floating-point representation for 0.01277 is 0.013.
04

d) 82,431

Since this number is given in a comma notation, our first step is to convert it into a regular number. 82,431 in regular notation is 82431. Now, we can represent it as 82431.000. After rounding it to three decimal places, the floating point representation stays the same, which is 82431.000.

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Most popular questions from this chapter

Let \\[ A=\left(\begin{array}{rr} 1 & 2 \\ -1 & -1 \end{array}\right) \quad \text { and } \quad \mathbf{u}_{0}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right) \\] (a) Compute \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3},\) and \(\mathbf{u}_{4},\) using the power method. (b) Explain why the power method will fail to converge in this case.

Suppose that \(A^{-1}\) and the \(L U\) factorization of \(A\) have already been determined. How many scalar additions and multiplications are necessary to compute \(A^{-1} \mathbf{b} ?\) Compare this number with the number of operations required to solve \(L U \mathbf{x}=\mathbf{b}\) using \(\mathrm{Al}\) gorithm \(7.2 .2 .\) Suppose that we have a number of systems to solve with the same coefficient matrix \(A .\) Is it worthwhile to compute \(A^{-1} ?\) Explain.

Let \(A=Q_{1} R_{1}=Q_{2} R_{2},\) where \(Q_{1}\) and \(Q_{2}\) are orthogonal and \(R_{1}\) and \(R_{2}\) are both upper triangular and nonsingular. (a) Show that \(Q_{1}^{T} Q_{2}\) is diagonal. (b) How do \(R_{1}\) and \(R_{2}\) compare? Explain.

Let \(A=X Y^{T}\), where \(X\) is an \(m \times r\) matrix, \(Y^{T}\) is an \(r \times n\) matrix, and \(X^{T} X\) and \(Y^{T} Y\) are both nonsingular. Show that the matrix \\[ B=Y\left(Y^{T} Y\right)^{-1}\left(X^{T} X\right)^{-1} X^{T} \\] satisfies the Penrose conditions and hence must equal \(A^{+} .\) Thus, \(A^{+}\) can be determined from any factorization of this form.

Let \(A\) be a nonsingular \(n \times n\) matrix and let \(Q\) be an \(n \times n\) orthogonal matrix. Show that (a) \(\operatorname{cond}_{2}(Q A)=\operatorname{cond}_{2}(A Q)=\operatorname{cond}_{2}(A)\) (b) if \(B=Q^{T} A Q,\) then \(\operatorname{cond}_{2}(B)=\operatorname{cond}_{2}(A)\)
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