(ㅡ) (6%) Please find a lower triangular matrix L and an upper triangular matrix U so that matrix A can be represented as a product of LU.

(二) (4%) Apply the result in (ㅡ) to solve the solution of Ax = b.

八(10%) Given， find u and y such that f(a)=u(x,y)+iv(x,y); determine where fis differentiable and where fis not.

九、(5%) Expand in Laurent series with center z = 3i in the neighborhood. Write down at least the first three terms. Determine the precise region of convergence.

(ㅡ) (5%) Find all zeros of cos z and show that all these are simple zeros.

(=) (5%) Evaluate where C:|2|=5,counter-clockwise. C COS Z

(a) Leave the design as it stands. Call this base machine Mbase. It has a clock rate of 500 MHz.

(b) Optimize the hardware. The hardware team claims that it can improve the processor design togive it a clock rate of 600 MHz. Call this machine Mopt.The following measurements have been made using a simulator for Mbase and Mopt.What are the CPI and MIPS (million instructions per second) rate for Mbase and Mopt? Copy Table1-1 to your answer sheet and fill in the blanks with your answers.

1. You are the lead designer of a new processor. The processor design and compiler arecomplete, and now you must decide whether to produce the current design as it stands or spendadditional time to improve it. Please consider and answer the following problems.1.2 The compiler team proposes to improve the compiler for the machine to further enhanceperformance. Call this combination of the improved compiler and the base machine Mcomp. Theinstruction improvements from this enhanced compiler have been estimated as follows.Instruction class Percentage of instruction executed vs. base machineFor example, if the base machine executed 500 class A instructions, Mcomp would execute 0.9 x500 = 450 class A instructions for the same program. What is the CPI for Mcomp? How much fasteris Mcomp than Mbase? Copy Table 1-2 to your answer sheet and fill in the blanks with your answers.

2. Here is a loop in C:Assume that A is an array of 100 elements and the variables g, h, i, and j are in registers $s1, $s2,$s3, and $s4, respectively. In addition, assume that the base of the array A is in $s5. Other registersthat can be used are $tO and $t1. A possible MISP assembly code corresponding to this C loop isshowed as follows:Please determine proper values for the operands (OPA, OPB, OPC, OPD, OPE). Copy the Table 2 toyour answer sheet and fill in the operand values.

3. Figure 1 shows a 16-bit adder consisting of four 4-bit ALUs using carry lookahead. Assume that gi = ai bi (generate) and pi = aitbi (propagate). Determine the value of P0, P1, P2, P3, G0, G1, G2, G3, and C4 (CarryOut) when adding two 16-bit numbers 0001 1010001 1001 1 and 1110010111101011 with Carry_ In bit equal to 0. Please copy Table 3 to your answer sheet and fill in the blanks with your answers.

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