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253
253
Index254
iv
Expression Templates
Expression templates give you an easy way to enter math expressions in standard
mathematical notation. When you insert a template, it appears on the entry line with
small blocks at positions where you can enter elements. A cursor shows which element
you can enter.
Position the cursor on each element, and type a value or expression for the element.
Fraction template
Note: See also / (divide), page 212.
Exponent template
Note: Type the first value, press l, and
then type the exponent. To return the cursor
to the baseline, press right arrow (¢).
Note: See also ^ (power), page 213.
Square root template
Note: See also √() (square root), page
223.
/p keys
Example:
l key
Example:
/q keys
Example:
Nth root template
Note: See also root(), page 154.
/l keys
Example:
Expression T emplates 1
Nth root template
/l keys
e exponent template
Natural exponential e raised to a power
Note: See also e^(), page 57.
Log template
Calculates log to a specified base. For a
default of base 10, omit the base.
Note: See also log(), page 107.
Piecewise template (2-piece)
Lets you create expressions and conditions
for a two-piece piecewise function. To add
a piece, click in the template and repeat the
template.
Note: See also piecewise(), page 133.
u keys
Example:
/s key
Example:
Catalog >
Example:
2 Expression Templates
Piecewise template (N-piece)
Lets you create expressions and conditions
for an N-piece piecewise function. Prompts
for N.
Note: See also piecewise(), page 133.
Catalog >
Example:
See the example for Piecewisetemplate (2piece).
System of 2 equations template
Creates a system of two equations. To add
a row to an existing system, click in the
template and repeat the template.
Note: See also system(), page 182.
System of N equations template
Lets you create a system of N equations.
Prompts for N.
Note: See also system(), page 182.
Absolute value template
Note: See also abs(), page 8.
Catalog >
Example:
Catalog >
Example:
See the example for Systemof equations
template (2-equation).
Catalog >
Example:
Expression T emplates 3
Absolute value template
Catalog >
dd°mm’ss.ss’’ template
Lets you enter angles in dd°mm’ss.ss’’
format, where dd is the number of decimal
degrees, mm is the number of minutes, and
ss.ss is the number of seconds.
Matrix template (2 x 2)
Creates a 2 x 2 matrix.
Matrix template (1 x 2)
.
Matrix template (2 x 1)
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Matrix template (m x n)
The template appears after you are
prompted to specify the number of rows
and columns.
4 Expression Templates
Catalog >
Example:
Matrix template (m x n)
Note: If you create a matrix with a large
number of rows and columns, it may take a
few moments to appear.
Catalog >
Sum template (Σ)
Note: See also Σ() (s umSeq), page 224.
Product template (Π)
Note: See also Π() (prodSeq), page 223.
First derivative template
The first derivative template can also be
used to calculate first derivative at a point.
Note: See also d() (derivative), page 221.
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Expression T emplates 5
Second derivative template
The second derivative template can also be
used to calculate second derivative at a
point.
Note: See also d() (derivative), page 221.
Catalog >
Example:
Nth derivative template
The nth derivative template can be used to
calculate the nth derivative.
Note: See also d() (derivative), page 221.
Definite integral template
Note: See also∫() integral(), page 221.
Indefinite integral template
Note: See also ∫() integral(), page 221.
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Limit template
6 Expression Templates
Catalog >
Example:
Limit template
Use − or (−) for left hand limit. Use + for
right hand limit.
Note: See also limit(), page 6.
Catalog >
Expression T emplates 7
Alphabetical Listing
Items whose names are not alphabetic (such as +, !, and >) are listed at the end of this
section, page 210. Unless otherwise specified, all examples in this section were
performed in the default reset mode, and all variables are assumed to be undefined.
A
abs()
abs(Expr1) ⇒ expression
abs(List1) ⇒ list
abs(Matrix1) ⇒ matrix
Returns the absolute value of the
argument.
Note: See also Absolute value template,
page 3.
If the argument is a complex number,
returns the number’s modulus.
Amortization function that returns a matrix
as an amortization table for a set of TVM
arguments.
NPmt is the number of payments to be
included in the table. The table starts with
the first payment.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM
arguments, page 195.
•If you omit Pmt, it defaults to
Pmt=tvmPmt
(N,I,PV,FV,PpY,CpY,PmtAt).
•If you omit FV, it defaults to FV=0.
•The defaults for PpY, CpY, and PmtAt
are the same as for the TVM functions.
Catalog >
Catalog >
8 Alphabetical Listing
amortTbl()
roundValue specifies the number of
decimal places for rounding. Default=2.
The columns in the result matrix are in this
order: Payment number, amount paid to
interest, amount paid to principal, and
balance.
The balance displayed in row n is the
balance after payment n.
You can use the output matrix as input for
the other amortization functions ΣInt() and
ΣPrn(), page 225, and bal(), page 17.
Catalog >
and
BooleanExpr1 and BooleanExpr2 ⇒
Boolean expression
BooleanList1 and BooleanList2 ⇒
Boolean list
BooleanMatrix1 and BooleanMatrix2 ⇒
Boolean matrix
Returns true or false or a simplified form of
the original entry.
Integer1 andInteger2 ⇒ integer
Compares two real integers bit-by-bit using
an and operation. Internally, both integers
are converted to signed, 64-bit binary
numbers. When corresponding bits are
compared, the result is 1 if both bits are 1;
otherwise, the result is 0. The returned
value represents the bit results, and is
displayed according to the Base mode.
You can enter the integers in any number
base. For a binary or hexadecimal entry, you
must use the 0b or 0h prefix, respectively.
Without a prefix, integers are treated as
decimal (base10).
Catalog >
In Hex basemode:
Important: Zero, not the letter O.
In Binbasemode:
In Dec base mode:
Note: A binary entry canhave upto 64 digits
(not countingthe0b prefix). A hexadecimal
entry canhave up to 16 digits.
Alphabetical Listing 9
angle()
angle(Expr1) ⇒ expression
Returns the angle of the argument,
interpreting the argument as a complex
number.
Note: All undefined variables are treated as
real variables.
angle(List1) ⇒ list
angle(Matrix1) ⇒ matrix
Returns a list or matrix of angles of the
elements in List1 or Matrix1, interpreting
each element as a complex number that
represents a two-dimensional rectangular
coordinate point.
Catalog >
In Degree angle mode:
In Gradianangle mode:
In Radian angle mode:
ANOVA
Catalog >
ANOVA List1,List2[,List3,...,List20][,Flag]
Performs a one-way analysis of variance for
comparing the means of two to 20
populations. A summary of results is stored
in the stat.results variable. (page 177)
Flag=0 for Data, Flag=1 for Stats
Output variableDescription
stat.FValueof the F statistic
stat.PValSmallest level of significance at which thenull hypothesis can be rejected
stat.dfDegrees of freedom of the groups
stat.SSSum of squaresof thegroups
stat.MSMean squares for the groups
10 A lphabetical Listing
Output variableDescription
stat.dfErrorDegrees of freedom of the errors
stat.SSErrorSum of squaresof theerrors
stat.MSErrorMean square for the errors
stat.spPooled standard deviation
stat.xbarlistMean of the inputof the lists
stat.CLowerList95%confidence intervals for the mean of each inputlist
stat.CUpperList95%confidenceintervals for the mean of each inputlist
ANOVA2way
Catalog >
ANOVA2way List1,List2[,List3,…,List10]
[,levRow]
Computes a two-way analysis of variance for
comparing the means of two to 10
populations. A summary of results is stored
in the stat.results variable. (See page 177.)
LevRow=0 for Block
LevRow=2,3,...,Len-1, for Two Factor,
where Len=length(List1)=length(List2) = …
= length(List10) and Len/LevRow Î
{2,3,…}
Outputs: Block Design
Output variableDescription
stat.FFstatistic of the columnfactor
stat.PValSmallest level of significance at which thenull hypothesis can be rejected
stat.PValInteractProbability valueof the interaction
stat.dfInteractDegrees of freedom of the interaction
stat.SSInteractSum of squaresof theinteraction
stat.MSInteractMean squares for interaction
ERROR Outputs
12 A lphabetical Listing
Output variableDescription
stat.dfErrorDegrees of freedom of the errors
stat.SSErrorSum of squaresof theerrors
stat.MSErrorMean squares for the errors
sStandard deviation of the error
Ans
Ans ⇒ value
Returns the result of the most recently
evaluated expression.
approx()
approx(Expr1) ⇒ expression
Returns the evaluation of the argument as
an expression containing decimal values,
when possible, regardless of the current
Auto or Approximate mode.
This is equivalent to entering the argument
and pressing /·.
approx(List1) ⇒ list
approx(Matrix1) ⇒ matrix
Returns a list or matrix where each
element has been evaluated to a decimal
value, when possible.
►approxFraction()
Expr►approxFraction([Tol]) ⇒
expression
/v keys
Catalog >
Catalog >
List►approxFraction([Tol]) ⇒ list
Matrix►approxFraction([Tol]) ⇒ matrix
Returns the input as a fraction, using a
tolerance of Tol. If Tol is omitted, a
tolerance of 5.E-14 is used.
Alphabetical Listing 13
►approxFraction()
Note: You can insert this function from the
computer keyboard by typing
@>approxFraction(...).
Catalog >
approxRational()
approxRational(Expr[, Tol]) ⇒ expression
approxRational(List[, Tol]) ⇒ list
approxRational(Matrix[, Tol]) ⇒ matrix
Returns the argument as a fraction using a
tolerance of Tol. If Tol is omitted, a
tolerance of 5.E-14 is used.
arccos()
arccosh()
arccot()
arccoth()
Catalog >
See cos⁻¹(), page 31.
See cosh⁻¹(), page 32.
See cot⁻¹(), page 33.
See coth⁻¹(), page 34.
arccsc()
arccsch()
14 A lphabetical Listing
See csc⁻¹(), page 37.
See csch⁻¹(), page 37.
arcLen()
arcLen(Expr1,Var,Start,End) ⇒
expression
Returns the arc length of Expr1 from
Start to End with respect to variable Var.
Arc length is calculated as an integral
assuming a function mode definition.
arcLen(List1,Var,Start,End) ⇒ list
Returns a list of the arc lengths of each
element of List1 from Start to End with
respect to Var.
Catalog >
arcsec()
arcsech()
arcsin()
arcsinh()
arctan()
arctanh()
augment()
augment(List1, List2) ⇒ list
See sec⁻¹(), page 158.
See sech⁻¹(), page 158.
See sin⁻¹(), page 168.
See sinh⁻¹(), page 169.
See tan⁻¹(), page 183.
See tanh⁻¹(), page 185.
Catalog >
Alphabetical Listing 15
augment()
Returns a new list that is List2 appended to
the end of List1.
augment(Matrix1, Matrix2) ⇒ matrix
Returns a new matrix that is Matrix2
appended to Matrix1. When the “,”
character is used, the matrices must have
equal row dimensions, and Matrix2 is
appended to Matrix1 as new columns.
Does not alter Matrix1 or Matrix2.
Catalog >
avgRC()
avgRC(Expr1, Var [=Value] [, Step]) ⇒
expression
avgRC(Expr1, Var [=Value] [, List1]) ⇒
list
avgRC(List1, Var [=Value] [, Step]) ⇒
list
avgRC(Matrix1, Var [=Value] [, Step]) ⇒
matrix
Returns the forward-difference quotient
(average rate of change).
Expr1 can be a user-defined function name
(see Func).
When Value is specified, it overrides any
prior variable assignment or any current “|”
substitution for the variable.
Step is the step value. If Step is omitted, it
defaults to 0.001.
Note that the similar function centralDiff()
uses the central-difference quotient.
Catalog >
16 A lphabetical Listing
B
bal()
bal(NPmt,N,I,PV ,[Pmt], [FV], [PpY],
[CpY], [PmtAt], [roundValue]) ⇒ value
bal(NPmt,amortTable) ⇒ value
Amortization function that calculates
schedule balance after a specified payment.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM
arguments, page 195.
NPmt specifies the payment number after
which you want the data calculated.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM
arguments, page 195.
•If you omit Pmt, it defaults to
Pmt=tvmPmt
(N,I,PV,FV,PpY,CpY,PmtAt).
•If you omit FV, it defaults to FV=0.
•The defaults for PpY, CpY, and PmtAt
are the same as for the TVM functions.
roundValue specifies the number of
decimal places for rounding. Default=2.
bal(NPmt,amortTable) calculates the
balance after payment number NPmt,
based on amortization table amortTable.
The amortTable argument must be a
matrix in the form described under
amortTbl(), page 8.
Note: See also ΣInt() and ΣPrn(), page 225.
Catalog >
►Base2
Integer1 ►Base2 ⇒ integer
Note: You can insert this operator from the
computer keyboard by typing @>Base2.
Catalog >
Alphabetical Listing 17
►Base2
Converts Integer1 to a binary number.
Binary or hexadecimal numbers always
have a 0b or 0h prefix, respectively. Use a
zero, not the letter O, followed by b or h.
0b binaryNumber
0h hexadecimalNumber
A binary number can have up to 64 digits. A
hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as
decimal (base10). The result is displayed in
binary, regardless of the Base mode.
Negative numbers are displayed in “two's
complement” form. For example,
⁻1is displayed as
0hFFFFFFFFFFFFFFFFin Hex base mode
0b111...111 (641’s)in Binary base mode
63
⁻2
is displayed as
0h8000000000000000in Hex base mode
0b100...000 (63 zeros)in Binary base mode
If you enter a decimal integer that is
outside the range of a signed, 64-bit binary
form, a symmetric modulo operation is
used to bring the value into the appropriate
range. Consider the following examples of
values outside the range.
263becomes ⁻263and is displayed as
0h8000000000000000in Hex base mode
0b100...000 (63 zeros)in Binary base mode
264becomes 0 and is displayed as
0h0in Hex base mode
0b0in Binary base mode
⁻263− 1 becomes 263− 1 and is displayed
as
0h7FFFFFFFFFFFFFFFin Hex base mode
0b111...111 (641’s)in Binary base mode
Catalog >
►Base10
Integer1 ►Base10 ⇒ integer
18 A lphabetical Listing
Catalog >
►Base10
Note: You can insert this operator from the
computer keyboard by typing @>Base10.
Converts Integer1 to a decimal (base10)
number. A binary or hexadecimal entry
must always have a 0b or 0h prefix,
respectively.
0b binaryNumber
0h hexadecimalNumber
Zero, not the letter O, followed by b or h.
A binary number can have up to 64 digits. A
hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as
decimal. The result is displayed in decimal,
regardless of the Base mode.
Catalog >
►Base16
Integer1 ►Base16 ⇒ integer
Note: You can insert this operator from the
computer keyboard by typing @>Base16.
Converts Integer1 to a hexadecimal
number. Binary or hexadecimal numbers
always have a 0b or 0h prefix, respectively.
0b binaryNumber
0h hexadecimalNumber
Zero, not the letter O, followed by b or h.
A binary number can have up to 64 digits. A
hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as
decimal (base10). The result is displayed in
hexadecimal, regardless of the Base mode.
If you enter a decimal integer that is too
large for a signed, 64-bit binary form, a
symmetric modulo operation is used to
bring the value into the appropriate range.
For more information, see ►Base2, page
17.
Catalog >
Alphabetical Listing 19
binomCdf()
binomCdf(n,p) ⇒ list
binomCdf(n,p,lowBound,upBound) ⇒
number if lowBound and upBound are
numbers, list if lowBound and upBound are
lists
binomCdf(n,p,upBound)for P(0≤X≤upBound)
⇒ number if upBound is a number, list if
upBound is a list
Computes a cumulative probability for the
discrete binomial distribution with n number
of trials and probability p of success on each
trial.
For P(X ≤ upBound), set lowBound=0
Catalog >
binomPdf()
binomPdf(n,p) ⇒ list
binomPdf(n,p,XVal) ⇒ number if XVal is a
number, list if XVal is a list
Computes a probability for the discrete
binomial distribution with n number of trials
and probability p of success on each trial.
C
ceiling(Expr1) ⇒ integer
Returns the nearest integer that is ≥ the
argument.
The argument can be a real or a complex
number.
Note: See also floor().
ceiling(List1) ⇒ list
ceiling(Matrix1) ⇒ matrix
Returns a list or matrix of the ceiling of
each element.
Catalog >
Catalog >
20 A lphabetical Listing
centralDiff()
centralDiff(Expr1,Var [=Value][,Step]) ⇒
expression
centralDiff(Expr1,Var [,Step])|Var=Value
⇒ expression
centralDiff(Expr1,Var [=Value][,List]) ⇒
list
centralDiff(List1,Var [=Value][,Step]) ⇒
list
centralDiff(Matrix1,Var [=Value][,Step])
⇒ matrix
Returns the numerical derivative using the
central difference quotient formula.
When Value is specified, it overrides any
prior variable assignment or any current “|”
substitution for the variable.
Step is the step value. If Step is omitted, it
defaults to 0.001.
When using List1 or Matrix1, the operation
gets mapped across the values in the list or
across the matrix elements.
Note: See also avgRC() and d().
Catalog >
cFactor()
cFactor(Expr1[,Var]) ⇒ expression
cFactor(List1[,Var]) ⇒ list
cFactor(Matrix1[,Var]) ⇒ matrix
cFactor(Expr1) returns Expr1 factored with
respect to all of its variables over a
common denominator.
Expr1 is factored as much as possible
toward linear rational factors even if this
introduces new non-real numbers. This
alternative is appropriate if you want
factorization with respect to more than one
variable.
Catalog >
Alphabetical Listing 21
cFactor()
cFactor(Expr1,Var) returns Expr1 factored
with respect to variable Var.
Expr1 is factored as much as possible
toward factors that are linear in Var, with
perhaps non-real constants, even if it
introduces irrational constants or
subexpressions that are irrational in other
variables.
The factors and their terms are sorted with
Var as the main variable. Similar powers of
Var are collected in each factor. Include
Var if factorization is needed with respect
to only that variable and you are willing to
accept irrational expressions in any other
variables to increase factorization with
respect to Var. There might be some
incidental factoring with respect to other
variables.
For the Auto setting of the Auto or
Approximate mode, including Var also
permits approximation with floating-point
coefficients where irrational coefficients
cannot be explicitly expressed concisely in
terms of the built-in functions. Even when
there is only one variable, including Var
might yield more complete factorization.
Note: See also factor().
Catalog >
To see the entire result, press £ and then
use¡and¢ to move thecursor.
char()
char(Integer) ⇒ character
Returns a character string containing the
character numbered Integer from the
handheld character set. The valid range for
Integer is 0–65535.
22 A lphabetical Listing
Catalog >
charPoly()
charPoly(squareMatrix,Var) ⇒
polynomial expression
charPoly(squareMatrix,Expr) ⇒
polynomial expression
charPoly(squareMatrix1,Matrix2) ⇒
polynomial expression
Returns the characteristic polynomial of
squareMatrix. The characteristic
polynomial of n×n matrix A, denoted by p
(λ), is the polynomial defined by
p
(λ) = det(λ•I−A)
A
where I denotes the n×n identity matrix.
squareMatrix1 and squareMatrix2 must
have the equal dimensions.
Catalog >
A
2
χ
2way
2
χ
2way obsMatrix
Catalog >
chi22way obsMatrix
Computes a χ2test for association on the
two-way table of counts in the observed
matrix obsMatrix. A summary of results is
stored in the stat.results variable. (page
177)
For information on the effect of empty
elements in a matrix, see “Empty (Void)
Elements,” page 236.
Output variableDescription
2
stat.χ
stat.PValSmallest level of significance at which thenull hypothesis can be rejected
stat.dfDegrees of freedom for the chi square statistics
stat.ExpMatMatrix of expectedelemental counttable, assuming nullhypothesis
stat.CompMatMatrix of elementalchi square statisticcontributions
Chi square stat: sum (observed - expected)2/expected
Alphabetical Listing 23
2
χ
Cdf()
2
χ
Cdf(lowBound,upBound,df) ⇒ number if
lowBound and upBound are numbers, list if
lowBound and upBound are lists
chi2Cdf(lowBound,upBound,df) ⇒ number
if lowBound and upBound are numbers, list
if lowBound and upBound are lists
Computes the χ2distribution probability
between lowBound and upBound for the
specified degrees of freedom df.
For P(X ≤ upBound), set lowBound = 0.
For information on the effect of empty
elements in a list, see “Empty (Void)
Elements,” page 236.
Catalog >
2
χ
GOF
2
χ
GOF obsList,expList,df
Catalog >
chi2GOF obsList,expList,df
Performs a test to confirm that sample data
is from a population that conforms to a
specified distribution. obsList is a list of
counts and must contain integers. A
summary of results is stored in the
stat.results variable. (See page 177.)
For information on the effect of empty
elements in a list, see “Empty (Void)
Elements,” page 236.
Output variableDescription
2
stat.χ
stat.PValSmallest level of significance at which thenull hypothesis can be rejected
stat.dfDegrees of freedom for the chi square statistics