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This chapter describes the Calc commands for doing simple calculations on numbers, such as addition, absolute value, and square roots. These commands work by removing the top one or two values from the stack, performing the desired operation, and pushing the result back onto the stack. If the operation cannot be performed, the result pushed is a formula instead of a number, such as `2/0' (because division by zero is illegal) or `sqrt(x)' (because the argument `x' is a formula).
Most of the commands described here can be invoked by a single keystroke. Some of the more obscure ones are two-letter sequences beginning with the f ("functions") prefix key.
See section 5.7 Numeric Prefix Arguments, for a discussion of the effect of numeric prefix arguments on commands in this chapter which do not otherwise interpret a prefix argument.
9.1 Basic Arithmetic | ||
9.2 Integer Truncation | ||
9.3 Complex Number Functions | ||
9.4 Conversions | ||
9.5 Date Arithmetic | ||
9.6 Financial Functions | ||
9.7 Binary Number Functions |
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The + (calc-plus
) command adds two numbers. The numbers may
be any of the standard Calc data types. The resulting sum is pushed back
onto the stack.
If both arguments of + are vectors or matrices (of matching dimensions), the result is a vector or matrix sum. If one argument is a vector and the other a scalar (i.e., a non-vector), the scalar is added to each of the elements of the vector to form a new vector. If the scalar is not a number, the operation is left in symbolic form: Suppose you added `x' to the vector `[1,2]'. You may want the result `[1+x,2+x]', or you may plan to substitute a 2-vector for `x' in the future. Since the Calculator can't tell which interpretation you want, it makes the safest assumption. See section 11.8 Reducing and Mapping Vectors, for a way to add `x' to every element of a vector.
If either argument of + is a complex number, the result will in general be complex. If one argument is in rectangular form and the other polar, the current Polar Mode determines the form of the result. If Symbolic Mode is enabled, the sum may be left as a formula if the necessary conversions for polar addition are non-trivial.
If both arguments of + are HMS forms, the forms are added according to the usual conventions of hours-minutes-seconds notation. If one argument is an HMS form and the other is a number, that number is converted from degrees or radians (depending on the current Angular Mode) to HMS format and then the two HMS forms are added.
If one argument of + is a date form, the other can be either a real number, which advances the date by a certain number of days, or an HMS form, which advances the date by a certain amount of time. Subtracting two date forms yields the number of days between them. Adding two date forms is meaningless, but Calc interprets it as the subtraction of one date form and the negative of the other. (The negative of a date form can be understood by remembering that dates are stored as the number of days before or after Jan 1, 1 AD.)
If both arguments of + are error forms, the result is an error form with an appropriately computed standard deviation. If one argument is an error form and the other is a number, the number is taken to have zero error. Error forms may have symbolic formulas as their mean and/or error parts; adding these will produce a symbolic error form result. However, adding an error form to a plain symbolic formula (as in `(a +/- b) + c') will not work, for the same reasons just mentioned for vectors. Instead you must write `(a +/- b) + (c +/- 0)'.
If both arguments of + are modulo forms with equal values of M, or if one argument is a modulo form and the other a plain number, the result is a modulo form which represents the sum, modulo M, of the two values.
If both arguments of + are intervals, the result is an interval which describes all possible sums of the possible input values. If one argument is a plain number, it is treated as the interval `[x .. x]'.
If one argument of + is an infinity and the other is not, the
result is that same infinity. If both arguments are infinite and in
the same direction, the result is the same infinity, but if they are
infinite in different directions the result is nan
.
The - (calc-minus
) command subtracts two values. The top
number on the stack is subtracted from the one behind it, so that the
computation 5 RET 2 - produces 3, not -3. All options
available for + are available for - as well.
The * (calc-times
) command multiplies two numbers. If one
argument is a vector and the other a scalar, the scalar is multiplied by
the elements of the vector to produce a new vector. If both arguments
are vectors, the interpretation depends on the dimensions of the
vectors: If both arguments are matrices, a matrix multiplication is
done. If one argument is a matrix and the other a plain vector, the
vector is interpreted as a row vector or column vector, whichever is
dimensionally correct. If both arguments are plain vectors, the result
is a single scalar number which is the dot product of the two vectors.
If one argument of * is an HMS form and the other a number, the HMS form is multiplied by that amount. It is an error to multiply two HMS forms together, or to attempt any multiplication involving date forms. Error forms, modulo forms, and intervals can be multiplied; see the comments for addition of those forms. When two error forms or intervals are multiplied they are considered to be statistically independent; thus, `[-2 .. 3] * [-2 .. 3]' is `[-6 .. 9]', whereas `[-2 .. 3] ^ 2' is `[0 .. 9]'.
The / (calc-divide
) command divides two numbers. When
dividing a scalar B by a square matrix A, the computation
performed is B times the inverse of A. This also occurs
if B is itself a vector or matrix, in which case the effect is
to solve the set of linear equations represented by B. If B
is a matrix with the same number of rows as A, or a plain vector
(which is interpreted here as a column vector), then the equation
A X = B is solved for the vector or matrix X. Otherwise,
if B is a non-square matrix with the same number of columns
as A, the equation X A = B is solved. If you wish a vector
B to be interpreted as a row vector to be solved as X A = B,
make it into a one-row matrix with C-u 1 v p first. To force a
left-handed solution with a square matrix B, transpose A and
B before dividing, then transpose the result.
HMS forms can be divided by real numbers or by other HMS forms. Error forms can be divided in any combination of ways. Modulo forms where both values and the modulo are integers can be divided to get an integer modulo form result. Intervals can be divided; dividing by an interval that encompasses zero or has zero as a limit will result in an infinite interval.
The ^ (calc-power
) command raises a number to a power. If
the power is an integer, an exact result is computed using repeated
multiplications. For non-integer powers, Calc uses Newton's method or
logarithms and exponentials. Square matrices can be raised to integer
powers. If either argument is an error (or interval or modulo) form,
the result is also an error (or interval or modulo) form.
If you press the I (inverse) key first, the I ^ command computes an Nth root: 125 RET 3 I ^ computes the number 5. (This is entirely equivalent to 125 RET 1:3 ^.)
The \ (calc-idiv
) command divides two numbers on the stack
to produce an integer result. It is equivalent to dividing with
/, then rounding down with F (calc-floor
), only a bit
more convenient and efficient. Also, since it is an all-integer
operation when the arguments are integers, it avoids problems that
/ F would have with floating-point roundoff.
The % (calc-mod
) command performs a "modulo" (or "remainder")
operation. Mathematically, `a%b = a - (a\b)*b', and is defined
for all real numbers a and b (except b=0). For
positive b, the result will always be between 0 (inclusive) and
b (exclusive). Modulo does not work for HMS forms and error forms.
If a is a modulo form, its modulo is changed to b, which
must be positive real number.
The : (calc-fdiv
) command [fdiv
function in a formula]
divides the two integers on the top of the stack to produce a fractional
result. This is a convenient shorthand for enabling Fraction Mode (with
m f) temporarily and using `/'. Note that during numeric entry
the : key is interpreted as a fraction separator, so to divide 8 by 6
you would have to type 8 RET 6 RET :. (Of course, in
this case, it would be much easier simply to enter the fraction directly
as 8:6 RET!)
The n (calc-change-sign
) command negates the number on the top
of the stack. It works on numbers, vectors and matrices, HMS forms, date
forms, error forms, intervals, and modulo forms.
The A (calc-abs
) [abs
] command computes the absolute
value of a number. The result of abs
is always a nonnegative
real number: With a complex argument, it computes the complex magnitude.
With a vector or matrix argument, it computes the Frobenius norm, i.e.,
the square root of the sum of the squares of the absolute values of the
elements. The absolute value of an error form is defined by replacing
the mean part with its absolute value and leaving the error part the same.
The absolute value of a modulo form is undefined. The absolute value of
an interval is defined in the obvious way.
The f A (calc-abssqr
) [abssqr
] command computes the
absolute value squared of a number, vector or matrix, or error form.
The f s (calc-sign
) [sign
] command returns 1 if its
argument is positive, -1 if its argument is negative, or 0 if its
argument is zero. In algebraic form, you can also write `sign(a,x)'
which evaluates to `x * sign(a)', i.e., either `x', `-x', or
zero depending on the sign of `a'.
The & (calc-inv
) [inv
] command computes the
reciprocal of a number, i.e., 1 / x. Operating on a square
matrix, it computes the inverse of that matrix.
The Q (calc-sqrt
) [sqrt
] command computes the square
root of a number. For a negative real argument, the result will be a
complex number whose form is determined by the current Polar Mode.
The f h (calc-hypot
) [hypot
] command computes the square
root of the sum of the squares of two numbers. That is, `hypot(a,b)'
is the length of the hypotenuse of a right triangle with sides a
and b. If the arguments are complex numbers, their squared
magnitudes are used.
The f Q (calc-isqrt
) [isqrt
] command computes the
integer square root of an integer. This is the true square root of the
number, rounded down to an integer. For example, `isqrt(10)'
produces 3. Note that, like \ [idiv
], this uses exact
integer arithmetic throughout to avoid roundoff problems. If the input
is a floating-point number or other non-integer value, this is exactly
the same as `floor(sqrt(x))'.
The f n (calc-min
) [min
] and f x (calc-max
)
[max
] commands take the minimum or maximum of two real numbers,
respectively. These commands also work on HMS forms, date forms,
intervals, and infinities. (In algebraic expressions, these functions
take any number of arguments and return the maximum or minimum among
all the arguments.)
The f M (calc-mant-part
) [mant
] function extracts
the "mantissa" part m of its floating-point argument; f X
(calc-xpon-part
) [xpon
] extracts the "exponent" part
e. The original number is equal to
m * 10^e,
where m is in the interval `[1.0 .. 10.0)' except that
m=e=0 if the original number is zero. For integers
and fractions, mant
returns the number unchanged and xpon
returns zero. The v u (calc-unpack
) command can also be
used to "unpack" a floating-point number; this produces an integer
mantissa and exponent, with the constraint that the mantissa is not
a multiple of ten (again except for the m=e=0 case).
The f S (calc-scale-float
) [scf
] function scales a number
by a given power of ten. Thus, `scf(mant(x), xpon(x)) = x' for any
real `x'. The second argument must be an integer, but the first
may actually be any numeric value. For example, `scf(5,-2) = 0.05'
or `1:20' depending on the current Fraction Mode.
The f [ (calc-decrement
) [decr
] and f ]
(calc-increment
) [incr
] functions decrease or increase
a number by one unit. For integers, the effect is obvious. For
floating-point numbers, the change is by one unit in the last place.
For example, incrementing `12.3456' when the current precision
is 6 digits yields `12.3457'. If the current precision had been
8 digits, the result would have been `12.345601'. Incrementing
`0.0' produces
10^-p, where p is the current
precision. These operations are defined only on integers and floats.
With numeric prefix arguments, they change the number by n units.
Note that incrementing followed by decrementing, or vice-versa, will almost but not quite always cancel out. Suppose the precision is 6 digits and the number `9.99999' is on the stack. Incrementing will produce `10.0000'; decrementing will produce `9.9999'. One digit has been dropped. This is an unavoidable consequence of the way floating-point numbers work.
Incrementing a date/time form adjusts it by a certain number of seconds. Incrementing a pure date form adjusts it by a certain number of days.
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There are four commands for truncating a real number to an integer, differing mainly in their treatment of negative numbers. All of these commands have the property that if the argument is an integer, the result is the same integer. An integer-valued floating-point argument is converted to integer form.
If you press H (calc-hyperbolic
) first, the result will be
expressed as an integer-valued floating-point number.
The F (calc-floor
) [floor
or ffloor
] command
truncates a real number to the next lower integer, i.e., toward minus
infinity. Thus 3.6 F produces 3, but _3.6 F produces
-4.
The I F (calc-ceiling
) [ceil
or fceil
]
command truncates toward positive infinity. Thus 3.6 I F produces
4, and _3.6 I F produces -3.
The R (calc-round
) [round
or fround
] command
rounds to the nearest integer. When the fractional part is .5 exactly,
this command rounds away from zero. (All other rounding in the
Calculator uses this convention as well.) Thus 3.5 R produces 4
but 3.4 R produces 3; _3.5 R produces -4.
The I R (calc-trunc
) [trunc
or ftrunc
]
command truncates toward zero. In other words, it "chops off"
everything after the decimal point. Thus 3.6 I R produces 3 and
_3.6 I R produces -3.
These functions may not be applied meaningfully to error forms, but they
do work for intervals. As a convenience, applying floor
to a
modulo form floors the value part of the form. Applied to a vector,
these functions operate on all elements of the vector one by one.
Applied to a date form, they operate on the internal numerical
representation of dates, converting a date/time form into a pure date.
There are two more rounding functions which can only be entered in
algebraic notation. The roundu
function is like round
except that it rounds up, toward plus infinity, when the fractional
part is .5. This distinction matters only for negative arguments.
Also, rounde
rounds to an even number in the case of a tie,
rounding up or down as necessary. For example, `rounde(3.5)' and
`rounde(4.5)' both return 4, but `rounde(5.5)' returns 6.
The advantage of round-to-even is that the net error due to rounding
after a long calculation tends to cancel out to zero. An important
subtle point here is that the number being fed to rounde
will
already have been rounded to the current precision before rounde
begins. For example, `rounde(2.500001)' with a current precision
of 6 will incorrectly, or at least surprisingly, yield 2 because the
argument will first have been rounded down to 2.5 (which
rounde
sees as an exact tie between 2 and 3).
Each of these functions, when written in algebraic formulas, allows a second argument which specifies the number of digits after the decimal point to keep. For example, `round(123.4567, 2)' will produce the answer 123.46, and `round(123.4567, -1)' will produce 120 (i.e., the cutoff is one digit to the left of the decimal point). A second argument of zero is equivalent to no second argument at all.
To compute the fractional part of a number (i.e., the amount which, when
added to `floor(N)', will produce N) just take N
modulo 1 using the %
command.
Note also the \ (integer quotient), f I (integer logarithm), and f Q (integer square root) commands, which are analogous to /, B, and Q, respectively, except that they take integer arguments and return the result rounded down to an integer.
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The J (calc-conj
) [conj
] command computes the
complex conjugate of a number. For complex number a+bi, the
complex conjugate is a-bi. If the argument is a real number,
this command leaves it the same. If the argument is a vector or matrix,
this command replaces each element by its complex conjugate.
The G (calc-argument
) [arg
] command computes the
"argument" or polar angle of a complex number. For a number in polar
notation, this is simply the second component of the pair
`(r;
theta)'.
The result is expressed according to the current angular mode and will
be in the range -180 degrees (exclusive) to +180 degrees
(inclusive), or the equivalent range in radians.
The calc-imaginary
command multiplies the number on the
top of the stack by the imaginary number i = (0,1). This
command is not normally bound to a key in Calc, but it is available
on the IMAG button in Keypad Mode.
The f r (calc-re
) [re
] command replaces a complex number
by its real part. This command has no effect on real numbers. (As an
added convenience, re
applied to a modulo form extracts
the value part.)
The f i (calc-im
) [im
] command replaces a complex number
by its imaginary part; real numbers are converted to zero. With a vector
or matrix argument, these functions operate element-wise.
The v p (calc-pack
) command can pack the top two numbers on
the stack into a composite object such as a complex number. With a
prefix argument of -1, it produces a rectangular complex number;
with an argument of -2, it produces a polar complex number.
(Also, see section 11.2 Building Vectors.)
The v u (calc-unpack
) command takes the complex number
(or other composite object) on the top of the stack and unpacks it
into its separate components.
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The commands described in this section convert numbers from one form to another; they are two-key sequences beginning with the letter c.
The c f (calc-float
) [pfloat
] command converts the
number on the top of the stack to floating-point form. For example,
23 is converted to 23.0, 3:2 is converted to
1.5, and 2.3 is left the same. If the value is a composite
object such as a complex number or vector, each of the components is
converted to floating-point. If the value is a formula, all numbers
in the formula are converted to floating-point. Note that depending
on the current floating-point precision, conversion to floating-point
format may lose information.
As a special exception, integers which appear as powers or subscripts are not floated by c f. If you really want to float a power, you can use a j s command to select the power followed by c f. Because c f cannot examine the formula outside of the selection, it does not notice that the thing being floated is a power. See section 12.1 Selecting Sub-Formulas.
The normal c f command is "pervasive" in the sense that it
applies to all numbers throughout the formula. The pfloat
algebraic function never stays around in a formula; `pfloat(a + 1)'
changes to `a + 1.0' as soon as it is evaluated.
With the Hyperbolic flag, H c f [float
] operates
only on the number or vector of numbers at the top level of its
argument. Thus, `float(1)' is 1.0, but `float(a + 1)'
is left unevaluated because its argument is not a number.
You should use H c f if you wish to guarantee that the final value, once all the variables have been assigned, is a float; you would use c f if you wish to do the conversion on the numbers that appear right now.
The c F (calc-fraction
) [pfrac
] command converts a
floating-point number into a fractional approximation. By default, it
produces a fraction whose decimal representation is the same as the
input number, to within the current precision. You can also give a
numeric prefix argument to specify a tolerance, either directly, or,
if the prefix argument is zero, by using the number on top of the stack
as the tolerance. If the tolerance is a positive integer, the fraction
is correct to within that many significant figures. If the tolerance is
a non-positive integer, it specifies how many digits fewer than the current
precision to use. If the tolerance is a floating-point number, the
fraction is correct to within that absolute amount.
The pfrac
function is pervasive, like pfloat
.
There is also a non-pervasive version, H c F [frac
],
which is analogous to H c f discussed above.
The c d (calc-to-degrees
) [deg
] command converts a
number into degrees form. The value on the top of the stack may be an
HMS form (interpreted as degrees-minutes-seconds), or a real number which
will be interpreted in radians regardless of the current angular mode.
The c r (calc-to-radians
) [rad
] command converts an
HMS form or angle in degrees into an angle in radians.
The c h (calc-to-hms
) [hms
] command converts a real
number, interpreted according to the current angular mode, to an HMS
form describing the same angle. In algebraic notation, the hms
function also accepts three arguments: `hms(h, m, s)'.
(The three-argument version is independent of the current angular mode.)
The calc-from-hms
command converts the HMS form on the top of the
stack into a real number according to the current angular mode.
The c p (calc-polar
) command converts the complex number on
the top of the stack from polar to rectangular form, or from rectangular
to polar form, whichever is appropriate. Real numbers are left the same.
This command is equivalent to the rect
or polar
functions in algebraic formulas, depending on the direction of
conversion. (It uses polar
, except that if the argument is
already a polar complex number, it uses rect
instead. The
I c p command always uses rect
.)
The c c (calc-clean
) [pclean
] command "cleans" the
number on the top of the stack. Floating point numbers are re-rounded
according to the current precision. Polar numbers whose angular
components have strayed from the -180 to +180 degree range
are normalized. (Note that results will be undesirable if the current
angular mode is different from the one under which the number was
produced!) Integers and fractions are generally unaffected by this
operation. Vectors and formulas are cleaned by cleaning each component
number (i.e., pervasively).
If the simplification mode is set below the default level, it is raised to the default level for the purposes of this command. Thus, c c applies the default simplifications even if their automatic application is disabled. See section 8.5 Simplification Modes.
A numeric prefix argument to c c sets the floating-point precision to that value for the duration of the command. A positive prefix (of at least 3) sets the precision to the specified value; a negative or zero prefix decreases the precision by the specified amount.
The keystroke sequences c 0 through c 9 are equivalent to c c with the corresponding negative prefix argument. If roundoff errors have changed 2.0 into 1.999999, typing c 1 to clip off one decimal place often conveniently does the trick.
The c c command with a numeric prefix argument, and the c 0 through c 9 commands, also "clip" very small floating-point numbers to zero. If the exponent is less than or equal to the negative of the specified precision, the number is changed to 0.0. For example, if the current precision is 12, then c 2 changes the vector `[1e-8, 1e-9, 1e-10, 1e-11]' to `[1e-8, 1e-9, 0, 0]'. Numbers this small generally arise from roundoff noise.
If the numbers you are using really are legitimately this small, you should avoid using the c 0 through c 9 commands. (The plain c c command rounds to the current precision but does not clip small numbers.)
One more property of c 0 through c 9, and of c c with a prefix argument, is that integer-valued floats are converted to plain integers, so that c 1 on `[1., 1.5, 2., 2.5, 3.]' produces `[1, 1.5, 2, 2.5, 3]'. This is not done for huge numbers (`1e100' is technically an integer-valued float, but you wouldn't want it automatically converted to a 100-digit integer).
With the Hyperbolic flag, H c c and H c 0 through H c 9
operate non-pervasively [clean
].
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The commands described in this section perform various conversions and calculations involving date forms (see section 6.9 Date Forms). They use the t (for time/date) prefix key followed by shifted letters.
The simplest date arithmetic is done using the regular + and - commands. In particular, adding a number to a date form advances the date form by a certain number of days; adding an HMS form to a date form advances the date by a certain amount of time; and subtracting two date forms produces a difference measured in days. The commands described here provide additional, more specialized operations on dates.
Many of these commands accept a numeric prefix argument; if you give plain C-u as the prefix, these commands will instead take the additional argument from the top of the stack.
9.5.1 Date Conversions | ||
9.5.2 Date Functions | ||
9.5.4 Time Zones | ||
9.5.3 Business Days |
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The t D (calc-date
) [date
] command converts a
date form into a number, measured in days since Jan 1, 1 AD. The
result will be an integer if date is a pure date form, or a
fraction or float if date is a date/time form. Or, if its
argument is a number, it converts this number into a date form.
With a numeric prefix argument, t D takes that many objects (up to six) from the top of the stack and interprets them in one of the following ways:
The `date(year, month, day)' function builds a pure date form out of the specified year, month, and day, which must all be integers. Year is a year number, such as 1991 (not the same as 91!). Month must be an integer in the range 1 to 12; day must be in the range 1 to 31. If the specified month has fewer than 31 days and day is too large, the equivalent day in the following month will be used.
The `date(month, day)' function builds a pure date form using the current year, as determined by the real-time clock.
The `date(year, month, day, hms)' function builds a date/time form using an hms form.
The `date(year, month, day, hour, minute, second)' function builds a date/time form. hour should be an integer in the range 0 to 23; minute should be an integer in the range 0 to 59; second should be any real number in the range `[0 .. 60)'. The last two arguments default to zero if omitted.
The t J (calc-julian
) [julian
] command converts
a date form into a Julian day count, which is the number of days
since noon on Jan 1, 4713 BC. A pure date is converted to an integer
Julian count representing noon of that day. A date/time form is
converted to an exact floating-point Julian count, adjusted to
interpret the date form in the current time zone but the Julian
day count in Greenwich Mean Time. A numeric prefix argument allows
you to specify the time zone; see section 9.5.4 Time Zones. Use a prefix of
zero to suppress the time zone adjustment. Note that pure date forms
are never time-zone adjusted.
This command can also do the opposite conversion, from a Julian day count (either an integer day, or a floating-point day and time in the GMT zone), into a pure date form or a date/time form in the current or specified time zone.
The t U (calc-unix-time
) [unixtime
] command
converts a date form into a Unix time value, which is the number of
seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
will be an integer if the current precision is 12 or less; for higher
precisions, the result may be a float with (precision-12)
digits after the decimal. Just as for t J, the numeric time
is interpreted in the GMT time zone and the date form is interpreted
in the current or specified zone. Some systems use Unix-like
numbering but with the local time zone; give a prefix of zero to
suppress the adjustment if so.
The t C (calc-convert-time-zones
) [tzconv
]
command converts a date form from one time zone to another. You
are prompted for each time zone name in turn; you can answer with
any suitable Calc time zone expression (see section 9.5.4 Time Zones).
If you answer either prompt with a blank line, the local time
zone is used for that prompt. You can also answer the first
prompt with $ to take the two time zone names from the
stack (and the date to be converted from the third stack level).
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The t N (calc-now
) [now
] command pushes the
current date and time on the stack as a date form. The time is
reported in terms of the specified time zone; with no numeric prefix
argument, t N reports for the current time zone.
The t P (calc-date-part
) command extracts one part
of a date form. The prefix argument specifies the part; with no
argument, this command prompts for a part code from 1 to 9.
The various part codes are described in the following paragraphs.
The M-1 t P [year
] function extracts the year number
from a date form as an integer, e.g., 1991. This and the
following functions will also accept a real number for an
argument, which is interpreted as a standard Calc day number.
Note that this function will never return zero, since the year
1 BC immediately precedes the year 1 AD.
The M-2 t P [month
] function extracts the month number
from a date form as an integer in the range 1 to 12.
The M-3 t P [day
] function extracts the day number
from a date form as an integer in the range 1 to 31.
The M-4 t P [hour
] function extracts the hour from
a date form as an integer in the range 0 (midnight) to 23. Note
that 24-hour time is always used. This returns zero for a pure
date form. This function (and the following two) also accept
HMS forms as input.
The M-5 t P [minute
] function extracts the minute
from a date form as an integer in the range 0 to 59.
The M-6 t P [second
] function extracts the second
from a date form. If the current precision is 12 or less,
the result is an integer in the range 0 to 59. For higher
precisions, the result may instead be a floating-point number.
The M-7 t P [weekday
] function extracts the weekday
number from a date form as an integer in the range 0 (Sunday)
to 6 (Saturday).
The M-8 t P [yearday
] function extracts the day-of-year
number from a date form as an integer in the range 1 (January 1)
to 366 (December 31 of a leap year).
The M-9 t P [time
] function extracts the time portion
of a date form as an HMS form. This returns `0@ 0' 0"'
for a pure date form.
The t M (calc-new-month
) [newmonth
] command
computes a new date form that represents the first day of the month
specified by the input date. The result is always a pure date
form; only the year and month numbers of the input are retained.
With a numeric prefix argument n in the range from 1 to 31,
t M computes the nth day of the month. (If n
is greater than the actual number of days in the month, or if
n is zero, the last day of the month is used.)
The t Y (calc-new-year
) [newyear
] command
computes a new pure date form that represents the first day of
the year specified by the input. The month, day, and time
of the input date form are lost. With a numeric prefix argument
n in the range from 1 to 366, t Y computes the
nth day of the year (366 is treated as 365 in non-leap
years). A prefix argument of 0 computes the last day of the
year (December 31). A negative prefix argument from -1 to
-12 computes the first day of the nth month of the year.
The t W (calc-new-week
) [newweek
] command
computes a new pure date form that represents the Sunday on or before
the input date. With a numeric prefix argument, it can be made to
use any day of the week as the starting day; the argument must be in
the range from 0 (Sunday) to 6 (Saturday). This function always
subtracts between 0 and 6 days from the input date.
Here's an example use of newweek
: Find the date of the next
Wednesday after a given date. Using M-3 t W or `newweek(d, 3)'
will give you the preceding Wednesday, so `newweek(d+7, 3)'
will give you the following Wednesday. A further look at the definition
of newweek
shows that if the input date is itself a Wednesday,
this formula will return the Wednesday one week in the future. An
exercise for the reader is to modify this formula to yield the same day
if the input is already a Wednesday. Another interesting exercise is
to preserve the time-of-day portion of the input (newweek
resets
the time to midnight; hint: how can newweek
be defined in terms
of the weekday
function?).
The `pwday(date)' function (not on any key) computes the
day-of-month number of the Sunday on or before date. With
two arguments, `pwday(date, day)' computes the day
number of the Sunday on or before day number day of the month
specified by date. The day must be in the range from
7 to 31; if the day number is greater than the actual number of days
in the month, the true number of days is used instead. Thus
`pwday(date, 7)' finds the first Sunday of the month, and
`pwday(date, 31)' finds the last Sunday of the month.
With a third weekday argument, pwday
can be made to look
for any day of the week instead of Sunday.
The t I (calc-inc-month
) [incmonth
] command
increases a date form by one month, or by an arbitrary number of
months specified by a numeric prefix argument. The time portion,
if any, of the date form stays the same. The day also stays the
same, except that if the new month has fewer days the day
number may be reduced to lie in the valid range. For example,
`incmonth(<Jan 31, 1991>)' produces `<Feb 28, 1991>'.
Because of this, t I t I and M-2 t I do not always give
the same results (`<Mar 28, 1991>' versus `<Mar 31, 1991>'
in this case).
The `incyear(date, step)' function increases a date form by the specified number of years, which may be any positive or negative integer. Note that `incyear(d, n)' is equivalent to `incmonth(d, 12*n)', but these do not have simple equivalents in terms of day arithmetic because months and years have varying lengths. If the step argument is omitted, 1 year is assumed. There is no keyboard command for this function; use C-u 12 t I instead.
There is no newday
function at all because F [floor
]
serves this purpose. Similarly, instead of incday
and
incweek
simply use d + n or d + 7 n.
See section 9.1 Basic Arithmetic, for the f ] [incr
] command
which can adjust a date/time form by a certain number of seconds.
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Often time is measured in "business days" or "working days," where weekends and holidays are skipped. Calc's normal date arithmetic functions use calendar days, so that subtracting two consecutive Mondays will yield a difference of 7 days. By contrast, subtracting two consecutive Mondays would yield 5 business days (assuming two-day weekends and the absence of holidays).
The t + (calc-business-days-plus
) [badd
]
and t - (calc-business-days-minus
) [bsub
]
commands perform arithmetic using business days. For t +,
one argument must be a date form and the other must be a real
number (positive or negative). If the number is not an integer,
then a certain amount of time is added as well as a number of
days; for example, adding 0.5 business days to a time in Friday
evening will produce a time in Monday morning. It is also
possible to add an HMS form; adding `12@ 0' 0"' also adds
half a business day. For t -, the arguments are either a
date form and a number or HMS form, or two date forms, in which
case the result is the number of business days between the two
dates.
By default, Calc considers any day that is not a Saturday or
Sunday to be a business day. You can define any number of
additional holidays by editing the variable Holidays
.
(There is an s H convenience command for editing this
variable.) Initially, Holidays
contains the vector
`[sat, sun]'. Entries in the Holidays
vector may
be any of the following kinds of objects:
sat
or sun
. This is really
a variable whose name is a three-letter, lower-case day name.
Holidays
vector, the default
`[1 .. 2737]' is used. (This is the absolute range of years
for which Calc's business-day algorithms will operate.)
Holidays
,
the full 24-hour day `[0 0' 0" .. 24 0' 0"]' is assumed.
(Regardless of the type of bounds you specify, the interval is
treated as inclusive on the low end and exclusive on the high end,
so that the work day goes from 9am up to, but not including, 5pm.)
If the Holidays
vector is empty, then t + and
t - will act just like + and - because there will
then be no difference between business days and calendar days.
Calc expands the intervals and formulas you give into a complete list of holidays for internal use. This is done mainly to make sure it can detect multiple holidays. (For example, `<Jan 1, 1989>' is both New Year's Day and a Sunday, but Calc's algorithms take care to count it only once when figuring the number of holidays between two dates.)
Since the complete list of holidays for all the years from 1 to 2737 would be huge, Calc actually computes only the part of the list between the smallest and largest years that have been involved in business-day calculations so far. Normally, you won't have to worry about this. Keep in mind, however, that if you do one calculation for 1992, and another for 1792, even if both involve only a small range of years, Calc will still work out all the holidays that fall in that 200-year span.
If you add a (positive) number of days to a date form that falls on a weekend or holiday, the date form is treated as if it were the most recent business day. (Thus adding one business day to a Friday, Saturday, or Sunday will all yield the following Monday.) If you subtract a number of days from a weekend or holiday, the date is effectively on the following business day. (So subtracting one business day from Saturday, Sunday, or Monday yields the preceding Friday.) The difference between two dates one or both of which fall on holidays equals the number of actual business days between them. These conventions are consistent in the sense that, if you add n business days to any date, the difference between the result and the original date will come out to n business days. (It can't be completely consistent though; a subtraction followed by an addition might come out a bit differently, since t + is incapable of producing a date that falls on a weekend or holiday.)
There is a holiday
function, not on any keys, that takes
any date form and returns 1 if that date falls on a weekend or
holiday, as defined in Holidays
, or 0 if the date is a
business day.
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Time zones and daylight savings time are a complicated business. The conversions to and from Julian and Unix-style dates automatically compute the correct time zone and daylight savings adjustment to use, provided they can figure out this information. This section describes Calc's time zone adjustment algorithm in detail, in case you want to do conversions in different time zones or in case Calc's algorithms can't determine the right correction to use.
Adjustments for time zones and daylight savings time are done by t U, t J, t N, and t C, but not by any other commands. In particular, `<may 1 1991> - <apr 1 1991>' evaluates to exactly 30 days even though there is a daylight-savings transition in between. This is also true for Julian pure dates: `julian(<may 1 1991>) - julian(<apr 1 1991>)'. But Julian and Unix date/times will adjust for daylight savings time: `julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)' evaluates to `29.95834' (that's 29 days and 23 hours) because one hour was lost when daylight savings commenced on April 7, 1991.
In brief, the idiom `julian(date1) - julian(date2)' computes the actual number of 24-hour periods between two dates, whereas `date1 - date2' computes the number of calendar days between two dates without taking daylight savings into account.
The calc-time-zone
[tzone
] command converts the time
zone specified by its numeric prefix argument into a number of
seconds difference from Greenwich mean time (GMT). If the argument
is a number, the result is simply that value multiplied by 3600.
Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
Daylight Savings time is in effect, one hour should be subtracted from
the normal difference.
If you give a prefix of plain C-u, calc-time-zone
(like other
date arithmetic commands that include a time zone argument) takes the
zone argument from the top of the stack. (In the case of t J
and t U, the normal argument is then taken from the second-to-top
stack position.) This allows you to give a non-integer time zone
adjustment. The time-zone argument can also be an HMS form, or
it can be a variable which is a time zone name in upper- or lower-case.
For example `tzone(PST) = tzone(8)' and `tzone(pdt) = tzone(7)'
(for Pacific standard and daylight savings times, respectively).
North American and European time zone names are defined as follows; note that for each time zone there is one name for standard time, another for daylight savings time, and a third for "generalized" time in which the daylight savings adjustment is computed from context.
YST PST MST CST EST AST NST GMT WET MET MEZ 9 8 7 6 5 4 3.5 0 -1 -2 -2 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ 8 7 6 5 4 3 2.5 -1 -2 -3 -3 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3 |
To define time zone names that do not appear in the above table,
you must modify the Lisp variable math-tzone-names
. This
is a list of lists describing the different time zone names; its
structure is best explained by an example. The three entries for
Pacific Time look like this:
( ( "PST" 8 0 ) ; Name as an upper-case string, then standard ( "PDT" 8 -1 ) ; adjustment, then daylight savings adjustment. ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone. |
With no arguments, calc-time-zone
or `tzone()' obtains an
argument from the Calc variable TimeZone
if a value has been
stored for that variable. If not, Calc runs the Unix `date'
command and looks for one of the above time zone names in the output;
if this does not succeed, `tzone()' leaves itself unevaluated.
The time zone name in the `date' output may be followed by a signed
adjustment, e.g., `GMT+5' or `GMT+0500' which specifies a
number of hours and minutes to be added to the base time zone.
Calc stores the time zone it finds into TimeZone
to speed
later calls to `tzone()'.
The special time zone name local
is equivalent to no argument,
i.e., it uses the local time zone as obtained from the date
command.
If the time zone name found is one of the standard or daylight
savings zone names from the above table, and Calc's internal
daylight savings algorithm says that time and zone are consistent
(e.g., PDT
accompanies a date that Calc's algorithm would also
consider to be daylight savings, or PST
accompanies a date
that Calc would consider to be standard time), then Calc substitutes
the corresponding generalized time zone (like PGT
).
If your system does not have a suitable `date' command, you
may wish to put a `(setq var-TimeZone ...)' in your Emacs
initialization file to set the time zone. The easiest way to do
this is to edit the TimeZone
variable using Calc's s T
command, then use the s p (calc-permanent-variable
)
command to save the value of TimeZone
permanently.
The t J and t U
commands with no numeric prefix
arguments do the same thing as `tzone()'. If the current
time zone is a generalized time zone, e.g., EGT
, Calc
examines the date being converted to tell whether to use standard
or daylight savings time. But if the current time zone is explicit,
e.g., EST
or EDT
, then that adjustment is used exactly
and Calc's daylight savings algorithm is not consulted.
Some places don't follow the usual rules for daylight savings time.
The state of Arizona, for example, does not observe daylight savings
time. If you run Calc during the winter season in Arizona, the
Unix date
command will report MST
time zone, which
Calc will change to MGT
. If you then convert a time that
lies in the summer months, Calc will apply an incorrect daylight
savings time adjustment. To avoid this, set your TimeZone
variable explicitly to MST
to force the use of standard,
non-daylight-savings time.
By default Calc always considers daylight savings time to begin at
2 a.m. on the first Sunday of April, and to end at 2 a.m. on the
last Sunday of October. This is the rule that has been in effect
in North America since 1987. If you are in a country that uses
different rules for computing daylight savings time, you have two
choices: Write your own daylight savings hook, or control time
zones explicitly by setting the TimeZone
variable and/or
always giving a time-zone argument for the conversion functions.
The Lisp variable math-daylight-savings-hook
holds the
name of a function that is used to compute the daylight savings
adjustment for a given date. The default is
math-std-daylight-savings
, which computes an adjustment
(either 0 or -1) using the North American rules given above.
The daylight savings hook function is called with four arguments:
The date, as a floating-point number in standard Calc format;
a six-element list of the date decomposed into year, month, day,
hour, minute, and second, respectively; a string which contains
the generalized time zone name in upper-case, e.g., "WEGT"
;
and a special adjustment to be applied to the hour value when
converting into a generalized time zone (see below).
The Lisp function math-prev-weekday-in-month
is useful for
daylight savings computations. This is an internal version of
the user-level pwday
function described in the previous
section. It takes four arguments: The floating-point date value,
the corresponding six-element date list, the day-of-month number,
and the weekday number (0-6).
The default daylight savings hook ignores the time zone name, but a more sophisticated hook could use different algorithms for different time zones. It would also be possible to use different algorithms depending on the year number, but the default hook always uses the algorithm for 1987 and later. Here is a listing of the default daylight savings hook:
(defun math-std-daylight-savings (date dt zone bump) (cond ((< (nth 1 dt) 4) 0) ((= (nth 1 dt) 4) (let ((sunday (math-prev-weekday-in-month date dt 7 0))) (cond ((< (nth 2 dt) sunday) 0) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 3 bump)) -1 0)) (t -1)))) ((< (nth 1 dt) 10) -1) ((= (nth 1 dt) 10) (let ((sunday (math-prev-weekday-in-month date dt 31 0))) (cond ((< (nth 2 dt) sunday) -1) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 2 bump)) 0 -1)) (t 0)))) (t 0)) ) |
The bump
parameter is equal to zero when Calc is converting
from a date form in a generalized time zone into a GMT date value.
It is -1 when Calc is converting in the other direction. The
adjustments shown above ensure that the conversion behaves correctly
and reasonably around the 2 a.m. transition in each direction.
There is a "missing" hour between 2 a.m. and 3 a.m. at the beginning of daylight savings time; converting a date/time form that falls in this hour results in a time value for the following hour, from 3 a.m. to 4 a.m. At the end of daylight savings time, the hour from 1 a.m. to 2 a.m. repeats itself; converting a date/time form that falls in this hour results in a time value for the first manifestion of that time (not the one that occurs one hour later).
If math-daylight-savings-hook
is nil
, then the
daylight savings adjustment is always taken to be zero.
In algebraic formulas, `tzone(zone, date)' computes the time zone adjustment for a given zone name at a given date. The date is ignored unless zone is a generalized time zone. If date is a date form, the daylight savings computation is applied to it as it appears. If date is a numeric date value, it is adjusted for the daylight-savings version of zone before being given to the daylight savings hook. This odd-sounding rule ensures that the daylight-savings computation is always done in local time, not in the GMT time that a numeric date is typically represented in.
The `dsadj(date, zone)' function computes the
daylight savings adjustment that is appropriate for date in
time zone zone. If zone is explicitly in or not in
daylight savings time (e.g., PDT
or PST
) the
date is ignored. If zone is a generalized time zone,
the algorithms described above are used. If zone is omitted,
the computation is done for the current time zone.
See section B. Reporting Bugs, for the address of Calc's author, if you
should wish to contribute your improved versions of
math-tzone-names
and math-daylight-savings-hook
to the Calc distribution.
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Calc's financial or business functions use the b prefix key followed by a shifted letter. (The b prefix followed by a lower-case letter is used for operations on binary numbers.)
Note that the rate and the number of intervals given to these functions must be on the same time scale, e.g., both months or both years. Mixing an annual interest rate with a time expressed in months will give you very wrong answers!
It is wise to compute these functions to a higher precision than you really need, just to make sure your answer is correct to the last penny; also, you may wish to check the definitions at the end of this section to make sure the functions have the meaning you expect.
9.6.1 Percentages | ||
9.6.2 Future Value | ||
9.6.3 Present Value | ||
9.6.4 Related Financial Functions | ||
9.6.5 Depreciation Functions | ||
9.6.6 Definitions |
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The M-% (calc-percent
) command takes a percentage value,
say 5.4, and converts it to an equivalent actual number. For example,
5.4 M-% enters 0.054 on the stack. (That's the META or
ESC key combined with %.)
Actually, M-% creates a formula of the form `5.4%'.
You can enter `5.4%' yourself during algebraic entry. The
`%' operator simply means, "the preceding value divided by
100." The `%' operator has very high precedence, so that
`1+8%' is interpreted as `1+(8%)', not as `(1+8)%'.
(The `%' operator is just a postfix notation for the
percent
function, just like `20!' is the notation for
`fact(20)', or twenty-factorial.)
The formula `5.4%' would normally evaluate immediately to 0.054, but the M-% command suppresses evaluation as it puts the formula onto the stack. However, the next Calc command that uses the formula `5.4%' will evaluate it as its first step. The net effect is that you get to look at `5.4%' on the stack, but Calc commands see it as `0.054', which is what they expect.
In particular, `5.4%' and `0.054' are suitable values for the rate arguments of the various financial functions, but the number `5.4' is probably not suitable--it represents a rate of 540 percent!
The key sequence M-% * effectively means "percent-of." For example, 68 RET 25 M-% * computes 17, which is 25% of 68 (and also 68% of 25, which comes out to the same thing).
The c % (calc-convert-percent
) command converts the
value on the top of the stack from numeric to percentage form.
For example, if 0.08 is on the stack, c % converts it to
`8%'. The quantity is the same, it's just represented
differently. (Contrast this with M-%, which would convert
this number to `0.08%'.) The = key is a convenient way
to convert a formula like `8%' back to numeric form, 0.08.
To compute what percentage one quantity is of another quantity, use / c %. For example, 17 RET 68 / c % displays `25%'.
The b % (calc-percent-change
) [relch
] command
calculates the percentage change from one number to another.
For example, 40 RET 50 b % produces the answer `25%',
since 50 is 25% larger than 40. A negative result represents a
decrease: 50 RET 40 b % produces `-20%', since 40 is
20% smaller than 50. (The answers are different in magnitude
because, in the first case, we're increasing by 25% of 40, but
in the second case, we're decreasing by 20% of 50.) The effect
of 40 RET 50 b % is to compute (50-40)/40, converting
the answer to percentage form as if by c %.
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The b F (calc-fin-fv
) [fv
] command computes
the future value of an investment. It takes three arguments
from the stack: `fv(rate, n, payment)'.
If you give payments of payment every year for n
years, and the money you have paid earns interest at rate per
year, then this function tells you what your investment would be
worth at the end of the period. (The actual interval doesn't
have to be years, as long as n and rate are expressed
in terms of the same intervals.) This function assumes payments
occur at the end of each interval.
The I b F [fvb
] command does the same computation,
but assuming your payments are at the beginning of each interval.
Suppose you plan to deposit $1000 per year in a savings account
earning 5.4% interest, starting right now. How much will be
in the account after five years? fvb(5.4%, 5, 1000) = 5870.73
.
Thus you will have earned $870 worth of interest over the years.
Using the stack, this calculation would have been
5.4 M-% 5 RET 1000 I b F. Note that the rate is expressed
as a number between 0 and 1, not as a percentage.
The H b F [fvl
] command computes the future value
of an initial lump sum investment. Suppose you could deposit
those five thousand dollars in the bank right now; how much would
they be worth in five years? fvl(5.4%, 5, 5000) = 6503.89
.
The algebraic functions fv
and fvb
accept an optional
fourth argument, which is used as an initial lump sum in the sense
of fvl
. In other words, fv(rate, n,
payment, initial) = fv(rate, n, payment)
+ fvl(rate, n, initial)
.
To illustrate the relationships between these functions, we could
do the fvb
calculation "by hand" using fvl
. The
final balance will be the sum of the contributions of our five
deposits at various times. The first deposit earns interest for
five years: fvl(5.4%, 5, 1000) = 1300.78
. The second
deposit only earns interest for four years: fvl(5.4%, 4, 1000) =
1234.13
. And so on down to the last deposit, which earns one
year's interest: fvl(5.4%, 1, 1000) = 1054.00
. The sum of
these five values is, sure enough, $5870.73, just as was computed
by fvb
directly.
What does fv(5.4%, 5, 1000) = 5569.96
mean? The payments
are now at the ends of the periods. The end of one year is the same
as the beginning of the next, so what this really means is that we've
lost the payment at year zero (which contributed $1300.78), but we're
now counting the payment at year five (which, since it didn't have
a chance to earn interest, counts as $1000). Indeed, 5569.96 =
5870.73 - 1300.78 + 1000 (give or take a bit of roundoff error).
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The b P (calc-fin-pv
) [pv
] command computes
the present value of an investment. Like fv
, it takes
three arguments: pv(rate, n, payment)
.
It computes the present value of a series of regular payments.
Suppose you have the chance to make an investment that will
pay $2000 per year over the next four years; as you receive
these payments you can put them in the bank at 9% interest.
You want to know whether it is better to make the investment, or
to keep the money in the bank where it earns 9% interest right
from the start. The calculation pv(9%, 4, 2000)
gives the
result 6479.44. If your initial investment must be less than this,
say, $6000, then the investment is worthwhile. But if you had to
put up $7000, then it would be better just to leave it in the bank.
Here is the interpretation of the result of pv
: You are
trying to compare the return from the investment you are
considering, which is fv(9%, 4, 2000) = 9146.26
, with
the return from leaving the money in the bank, which is
fvl(9%, 4, x)
where x is the amount of money
you would have to put up in advance. The pv
function
finds the break-even point, x = 6479.44, at which
fvl(9%, 4, 6479.44)
is also equal to 9146.26. This is
the largest amount you should be willing to invest.
The I b P [pvb
] command solves the same problem,
but with payments occurring at the beginning of each interval.
It has the same relationship to fvb
as pv
has
to fv
. For example pvb(9%, 4, 2000) = 7062.59
,
a larger number than pv
produced because we get to start
earning interest on the return from our investment sooner.
The H b P [pvl
] command computes the present value of
an investment that will pay off in one lump sum at the end of the
period. For example, if we get our $8000 all at the end of the
four years, pvl(9%, 4, 8000) = 5667.40
. This is much
less than pv
reported, because we don't earn any interest
on the return from this investment. Note that pvl
and
fvl
are simple inverses: fvl(9%, 4, 5667.40) = 8000
.
You can give an optional fourth lump-sum argument to pv
and pvb
; this is handled in exactly the same way as the
fourth argument for fv
and fvb
.
The b N (calc-fin-npv
) [npv
] command computes
the net present value of a series of irregular investments.
The first argument is the interest rate. The second argument is
a vector which represents the expected return from the investment
at the end of each interval. For example, if the rate represents
a yearly interest rate, then the vector elements are the return
from the first year, second year, and so on.
Thus, npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44
.
Obviously this function is more interesting when the payments are
not all the same!
The npv
function can actually have two or more arguments.
Multiple arguments are interpreted in the same way as for the
vector statistical functions like vsum
.
See section 11.7.1 Single-Variable Statistics. Basically, if there are several
payment arguments, each either a vector or a plain number, all these
values are collected left-to-right into the complete list of payments.
A numeric prefix argument on the b N command says how many
payment values or vectors to take from the stack.
The I b N [npvb
] command computes the net present
value where payments occur at the beginning of each interval
rather than at the end.
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The functions in this section are basically inverses of the present value functions with respect to the various arguments.
The b M (calc-fin-pmt
) [pmt
] command computes
the amount of periodic payment necessary to amortize a loan.
Thus pmt(rate, n, amount)
equals the
value of payment such that pv(rate, n,
payment) = amount
.
The I b M [pmtb
] command does the same computation
but using pvb
instead of pv
. Like pv
and
pvb
, these functions can also take a fourth argument which
represents an initial lump-sum investment.
The H b M key just invokes the fvl
function, which is
the inverse of pvl
. There is no explicit pmtl
function.
The b # (calc-fin-nper
) [nper
] command computes
the number of regular payments necessary to amortize a loan.
Thus nper(rate, payment, amount)
equals
the value of n such that pv(rate, n,
payment) = amount
. If payment is too small
ever to amortize a loan for amount at interest rate rate,
the nper
function is left in symbolic form.
The I b # [nperb
] command does the same computation
but using pvb
instead of pv
. You can give a fourth
lump-sum argument to these functions, but the computation will be
rather slow in the four-argument case.
The H b # [nperl
] command does the same computation
using pvl
. By exchanging payment and amount you
can also get the solution for fvl
. For example,
nperl(8%, 2000, 1000) = 9.006
, so if you place $1000 in a
bank account earning 8%, it will take nine years to grow to $2000.
The b T (calc-fin-rate
) [rate
] command computes
the rate of return on an investment. This is also an inverse of pv
:
rate(n, payment, amount)
computes the value of
rate such that pv(rate, n, payment) =
amount
. The result is expressed as a formula like `6.3%'.
The I b T [rateb
] and H b T [ratel
]
commands solve the analogous equations with pvb
or pvl
in place of pv
. Also, rate
and rateb
can
accept an optional fourth argument just like pv
and pvb
.
To redo the above example from a different perspective,
ratel(9, 2000, 1000) = 8.00597%
, which says you will need an
interest rate of 8% in order to double your account in nine years.
The b I (calc-fin-irr
) [irr
] command is the
analogous function to rate
but for net present value.
Its argument is a vector of payments. Thus irr(payments)
computes the rate such that npv(rate, payments) = 0
;
this rate is known as the internal rate of return.
The I b I [irrb
] command computes the internal rate of
return assuming payments occur at the beginning of each period.
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The functions in this section calculate depreciation, which is the amount of value that a possession loses over time. These functions are characterized by three parameters: cost, the original cost of the asset; salvage, the value the asset will have at the end of its expected "useful life"; and life, the number of years (or other periods) of the expected useful life.
There are several methods for calculating depreciation that differ in the way they spread the depreciation over the lifetime of the asset.
The b S (calc-fin-sln
) [sln
] command computes the
"straight-line" depreciation. In this method, the asset depreciates
by the same amount every year (or period). For example,
`sln(12000, 2000, 5)' returns 2000. The asset costs $12000
initially and will be worth $2000 after five years; it loses $2000
per year.
The b Y (calc-fin-syd
) [syd
] command computes the
accelerated "sum-of-years'-digits" depreciation. Here the depreciation
is higher during the early years of the asset's life. Since the
depreciation is different each year, b Y takes a fourth period
parameter which specifies which year is requested, from 1 to life.
If period is outside this range, the syd
function will
return zero.
The b D (calc-fin-ddb
) [ddb
] command computes an
accelerated depreciation using the double-declining balance method.
It also takes a fourth period parameter.
For symmetry, the sln
function will accept a period
parameter as well, although it will ignore its value except that the
return value will as usual be zero if period is out of range.
For example, pushing the vector [1,2,3,4,5] (perhaps with v x 5) and then mapping V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$), ddb(12000,2000,5,$)] RET produces a matrix that allows us to compare the three depreciation methods:
[ [ 2000, 3333, 4800 ] [ 2000, 2667, 2880 ] [ 2000, 2000, 1728 ] [ 2000, 1333, 592 ] [ 2000, 667, 0 ] ] |
(Values have been rounded to nearest integers in this figure.)
We see that sln
depreciates by the same amount each year,
syd depreciates more at the beginning and less at the end,
and ddb weights the depreciation even more toward the beginning.
Summing columns with V R : + yields [10000, 10000, 10000]; the total depreciation in any method is (by definition) the difference between the cost and the salvage value.
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For your reference, here are the actual formulas used to compute Calc's financial functions.
Calc will not evaluate a financial function unless the rate or
n argument is known. However, payment or amount can
be a variable. Calc expands these functions according to the
formulas below for symbolic arguments only when you use the a "
(calc-expand-formula
) command, or when taking derivatives or
integrals or solving equations involving the functions.
These formulas are shown using the conventions of "Big" display
mode (d B); for example, the formula for fv
written
linearly is `pmt * ((1 + rate)^n) - 1) / rate'.
n (1 + rate) - 1 fv(rate, n, pmt) = pmt * --------------- rate n ((1 + rate) - 1) (1 + rate) fvb(rate, n, pmt) = pmt * ---------------------------- rate n fvl(rate, n, pmt) = pmt * (1 + rate) -n 1 - (1 + rate) pv(rate, n, pmt) = pmt * ---------------- rate -n (1 - (1 + rate) ) (1 + rate) pvb(rate, n, pmt) = pmt * ----------------------------- rate -n pvl(rate, n, pmt) = pmt * (1 + rate) -1 -2 -3 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate) -1 -2 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate) -n (amt - x * (1 + rate) ) * rate pmt(rate, n, amt, x) = ------------------------------- -n 1 - (1 + rate) -n (amt - x * (1 + rate) ) * rate pmtb(rate, n, amt, x) = ------------------------------- -n (1 - (1 + rate) ) (1 + rate) amt * rate nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate) pmt amt * rate nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate) pmt * (1 + rate) amt nperl(rate, pmt, amt) = - log(---, 1 + rate) pmt 1/n pmt ratel(n, pmt, amt) = ------ - 1 1/n amt cost - salv sln(cost, salv, life) = ----------- life (cost - salv) * (life - per + 1) syd(cost, salv, life, per) = -------------------------------- life * (life + 1) / 2 book * 2 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far life |
In pmt
and pmtb
, x=0 if omitted.
These functions accept any numeric objects, including error forms, intervals, and even (though not very usefully) complex numbers. The above formulas specify exactly the behavior of these functions with all sorts of inputs.
Note that if the first argument to the log
in nper
is
negative, nper
leaves itself in symbolic form rather than
returning a (financially meaningless) complex number.
`rate(num, pmt, amt)' solves the equation
`pv(rate, num, pmt) = amt' for `rate' using H a R
(calc-find-root
), with the interval `[.01% .. 100%]'
for an initial guess. The rateb
function is the same except
that it uses pvb
. Note that ratel
can be solved
directly; its formula is shown in the above list.
Similarly, `irr(pmts)' solves the equation `npv(rate, pmts) = 0' for `rate'.
If you give a fourth argument to nper
or nperb
, Calc
will also use H a R to solve the equation using an initial
guess interval of `[0 .. 100]'.
A fourth argument to fv
simply sums the two components
calculated from the above formulas for fv
and fvl
.
The same is true of fvb
, pv
, and pvb
.
The ddb function is computed iteratively; the "book" value
starts out equal to cost, and decreases according to the above
formula for the specified number of periods. If the book value
would decrease below salvage, it only decreases to salvage
and the depreciation is zero for all subsequent periods. The ddb
function returns the amount the book value decreased in the specified
period.
The Calc financial function names were borrowed mostly from Microsoft
Excel and Borland's Quattro. The ratel
function corresponds to
`@CGR' in Borland's Reflex. The nper
and nperl
functions correspond to `@TERM' and `@CTERM' in Quattro,
respectively. Beware that the Calc functions may take their arguments
in a different order than the corresponding functions in your favorite
spreadsheet.
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The commands in this chapter all use two-letter sequences beginning with the b prefix.
The "binary" operations actually work regardless of the currently displayed radix, although their results make the most sense in a radix like 2, 8, or 16 (as obtained by the d 2, d 8, or d 6 commands, respectively). You may also wish to enable display of leading zeros with d z. See section 8.7.1 Radix Modes.
The Calculator maintains a current word size w, an arbitrary positive or negative integer. For a positive word size, all of the binary operations described here operate modulo 2^w. In particular, negative arguments are converted to positive integers modulo 2^w by all binary functions.
If the word size is negative, binary operations produce 2's complement integers from -(2^(-w-1)) to 2^(-w-1)-1 inclusive. Either mode accepts inputs in any range; the sign of w affects only the results produced.
The b c (calc-clip
)
[clip
] command can be used to clip a number by reducing it modulo
2^w. The commands described in this chapter automatically clip
their results to the current word size. Note that other operations like
addition do not use the current word size, since integer addition
generally is not "binary." (However, see section 8.5 Simplification Modes,
calc-bin-simplify-mode
.) For example, with a word size of 8
bits b c converts a number to the range 0 to 255; with a word
size of -8 b c converts to the range -128 to 127.
The default word size is 32 bits. All operations except the shifts and
rotates allow you to specify a different word size for that one
operation by giving a numeric prefix argument: C-u 8 b c clips the
top of stack to the range 0 to 255 regardless of the current word size.
To set the word size permanently, use b w (calc-word-size
).
This command displays a prompt with the current word size; press RET
immediately to keep this word size, or type a new word size at the prompt.
When the binary operations are written in symbolic form, they take an optional second (or third) word-size parameter. When a formula like `and(a,b)' is finally evaluated, the word size current at that time will be used, but when `and(a,b,-8)' is evaluated, a word size of -8 will always be used. A symbolic binary function will be left in symbolic form unless the all of its argument(s) are integers or integer-valued floats.
If either or both arguments are modulo forms for which M is a power of two, that power of two is taken as the word size unless a numeric prefix argument overrides it. The current word size is never consulted when modulo-power-of-two forms are involved.
The b a (calc-and
) [and
] command computes the bitwise
AND of the two numbers on the top of the stack. In other words, for each
of the w binary digits of the two numbers (pairwise), the corresponding
bit of the result is 1 if and only if both input bits are 1:
`and(2#1100, 2#1010) = 2#1000'.
The b o (calc-or
) [or
] command computes the bitwise
inclusive OR of two numbers. A bit is 1 if either of the input bits, or
both, are 1: `or(2#1100, 2#1010) = 2#1110'.
The b x (calc-xor
) [xor
] command computes the bitwise
exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
is 1: `xor(2#1100, 2#1010) = 2#0110'.
The b d (calc-diff
) [diff
] command computes the bitwise
difference of two numbers; this is defined by `diff(a,b) = and(a,not(b))',
so that `diff(2#1100, 2#1010) = 2#0100'.
The b n (calc-not
) [not
] command computes the bitwise
NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
The b l (calc-lshift-binary
) [lsh
] command shifts a
number left by one bit, or by the number of bits specified in the numeric
prefix argument. A negative prefix argument performs a logical right shift,
in which zeros are shifted in on the left. In symbolic form, `lsh(a)'
is short for `lsh(a,1)', which in turn is short for `lsh(a,n,w)'.
Bits shifted "off the end," according to the current word size, are lost.
The H b l command also does a left shift, but it takes two arguments from the stack (the value to shift, and, at top-of-stack, the number of bits to shift). This version interprets the prefix argument just like the regular binary operations, i.e., as a word size. The Hyperbolic flag has a similar effect on the rest of the binary shift and rotate commands.
The b r (calc-rshift-binary
) [rsh
] command shifts a
number right by one bit, or by the number of bits specified in the numeric
prefix argument: `rsh(a,n) = lsh(a,-n)'.
The b L (calc-lshift-arith
) [ash
] command shifts a
number left. It is analogous to lsh
, except that if the shift
is rightward (the prefix argument is negative), an arithmetic shift
is performed as described below.
The b R (calc-rshift-arith
) [rash
] command performs
an "arithmetic" shift to the right, in which the leftmost bit (according
to the current word size) is duplicated rather than shifting in zeros.
This corresponds to dividing by a power of two where the input is interpreted
as a signed, twos-complement number. (The distinction between the `rsh'
and `rash' operations is totally independent from whether the word
size is positive or negative.) With a negative prefix argument, this
performs a standard left shift.
The b t (calc-rotate-binary
) [rot
] command rotates a
number one bit to the left. The leftmost bit (according to the current
word size) is dropped off the left and shifted in on the right. With a
numeric prefix argument, the number is rotated that many bits to the left
or right.
See section 11.6 Set Operations using Vectors, for the b p and b u commands that pack and unpack binary integers into sets. (For example, b u unpacks the number `2#11001' to the set of bit-numbers `[0, 3, 4]'.) Type b u V # to count the number of "1" bits in a binary integer.
Another interesting use of the set representation of binary integers is to reverse the bits in, say, a 32-bit integer. Type b u to unpack; type 31 TAB - to replace each bit-number in the set with 31 minus that bit-number; type b p to pack the set back into a binary integer.
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