Marvin L. Stein

Changing from Analog to Digital Programming by Digital Techniques

A quick and economical way of writing digital computer programs for the solution of problems already set up for analog computation would be welcomed by analog computer users. They could then use a digital computer to check analog solutions or to obtain greater accuracy for selected analog runs. The method which the authors advocate for accomplishing the shift from analog to digital programming consists in having the digital computer itself analyze a description of the analog setup diagram in order to deduce the differential equations being solved and compile a complete program including input and output for the integration of these equations. This paper is devoted to the first part of this process--the deduction by a digital computer of the differential equations which an analog computer will solve directly from a description of its setup diagram. In the paper, theory and techniques are developed which permit the analysis of setup diagrams representing systems of differential equations of the form

Automatic Digital Programming of Analog Computers

A digital technique for manipulating symbols defining a mathematical expression is introduced and it is shown how this technique can be applied so that a digital computer can automatically compile an analog computer program directly from a mathematical description of the problem to be solved. The mathematical problem considered is that of solving a system of nonlinear first-order differential equations with initial conditions. It is assumed that these equations have been prescaled and are expressed in terms of the physical variables rather than the original mathematical variables. A resume of a method for reducing the original symbol sequence defining the differential equations to symbol subsequences from which the analog program can be formed is given. It is then shown how the analog program is to be generated from these symbol sequences. To illustrate the detailed application of the technique an analog program is compiled for a particular system of differential equations.

Scaling Machine Arithmetic

A general approach to the problem of scaling machine arithmetic is developed. This leads to the determination of inequalities that can serve as a basis for the derivation of systematic scaling techniques. The inequalities and techniques are shown to apply to complement arithmetic with either integral or fractional machine operations and to absolute value and sign arithmetic for both types of operations. A detailed discussion is presented for the case of complement integer arithmetic. The connections with floating point arithmetic are derived.

On complement division

The division algorithm theorem is expressed in a form that permits it to serve as the basis for devising division operations that produce both quotient and remainder in complement form. Algorithms for division yielding complement results are derived for numbers represented in any base greater than one. Both radix and radix-less-one complementation schemes are considered. The binary form of the algorithms thus includes both twos and ones complement implementation. The problem of quotient overflow for complement results is dealt with as is that of selecting an appropriate form of the remainder condition for complement division.

Sorting Implicit Outputs in Digital Simulation

The Stein-Rose sorting algorithm for digital simulation is reexamined and reformulated in current terms. It is pointed out that the algorithm already contains within itself the means for the automatic detection of feedback loops (implicit outputs) in analog programs. The algorithm is then extended so that it can handle implicit outputs and the extended version is shown to be finite. Thus compilers for digital simulation languages utilizing the extended algorithm need not require that programmers take a priori action either to remove feedback loops from analog programs or to identify such loops to the compiler.

Fixed and Floating Point Arithmetic; Scaling

This chapter elaborates describes fixed and floating point methods of computing and some of the arithmetic operations related to them. Fixed-point operations in a digital computer treat all sequences of digits representing arithmetic operands as though an imaginary base point were in a fixed position. Shifting operations are designed to move the digits in a register either to the right or to the left of their initial position. The CDC 1604 left shift instructions are end around or circular. The multiply integer and divide integer operations treat sequences of digits as though the binary point were at the right-hand end of the sequence. In a machine using integral fixed point operations, all arithmetic operations are carried out on integers with the base point considered to be at the right-hand end of each sequence of digits. If the sum or difference is formed in an arithmetic register of extra length, the arithmetic register may accommodate it but an ordinary storage register may not. The direct application of fixed-point arithmetic instructions to floating point machine operands will produce incorrect results.

Assembly of Complete Programs

This chapter discusses the assembly of complete programs. Many assembly programs that go beyond the regional in their capabilities often prepare an intermediate version, in regional form, of the programs they are putting together. The SURAP type of assembly program employs a format that gives the region tag in numerical form. Each program for assembly must be initiated by a Program Name statement and be terminated with an End statement. The subroutine INTERP is entered by a return jump to the symbolic address INTERP from the upper instruction of a program step in the main program. The INTERP program itself is a subroutine of the program from which it draws the interpretive instructions that it interprets and executes.

CHAPTER 3 – Elementary Coding

Publisher Summary This chapter focuses on elementary coding, which represents the number of instructions that are arranged into meaningful patterns in such a way that the computer is made to perform a given calculation. The coding technique that should ordinarily be employed consists of creating the instructions for handling each item of data as it is needed from a small set of prototype instructions. An iterative loop of addresses consists of a sequence of addresses that the control consults repetitively in the execution of a code. If a program causes the computer to carry out on the same data exactly the same calculation every time it is executed, its initial address will be called a proper entry point for the program. The coder should not blindly set out to write a program without working out a precise plan of attack beforehand. An index register to index register transmission can be carried out by means of an appropriate combination of the store index and load index or enter index instructions.

CHAPTER 5 – Nonarithmetic Operations

Publisher Summary This chapter focuses on certain aspects of nonarithmetic operations. The stop instructions in the CDC 1604 operate as unconditional jump instructions. In explicit computers, each instruction includes a jump, that is, each instruction contains the location of the instruction which is to follow. An instruction such as the sign jump specifically tests the single sign bit. In a computer, the basic operation of forming a logical product or sum is usually utilized in a number of related operations giving specialized combinations of logical and transmissive, complementing, and arithmetic instructions. In the CDC 1604, there are several instructions that facilitate the problem of dealing with individual bits in a register. A basic problem that often arises in programming is that of selecting a quantity with a given property from a set of stored data. The names equality search and threshold search reflect the fact that the basic conditions are the same.

CHAPTER 2 – Machine Organization

Publisher Summary This chapter discusses the way in which computing machines are organized to perform their functions. The number of digits that constitute the content of a register defines the length of that register. In the CDC 1604, all storage registers are 48 bits in length. The word in a storage register, when interpreted as an arithmetic operand, is usually considered to be either a fraction or an integer, in which case the base point is thought of as fixed or it is thought of as a combination of the two. The accumulator is utilized in almost all of the arithmetic instructions of the computer. If a sequence of digits is to be interpreted as an instruction, it must contain minimally a means of specifying the particular operation to be performed. If operands are involved, they must also be specified. The CDC 1604 instruction word is made up of 24 bits. Because instructions and operands occupy the same set of storage registers, they are indistinguishable. The amount of peripheral input–output equipment used by a CDC 1604 varies in different installations.