# gmtmath

Reverse Polish Notation (RPN) calculator for data tables

## Synopsis

gmt math [ -A t_f(t)[+e][+r][+s|w] ] [ -C cols ] [ -E eigen ] [ -I ] [ -N n_col[/t_col] ] [ -Q[c|i|p|n] ] [ -S[f|l] ] [ -T[min/max/inc[+b|i|l|n]|file|list] ] [ -V[level] ] [ -b binary ] [ -d nodata[+c col] ] [ -e regexp ] [ -f flags ] [ -g gaps ] [ -h headers ] [ -i flags ] [ -o flags ] [ -q flags ] [ -s flags ] [ -w flags ] [ --PAR=value ] operand [ operand ] OPERATOR [ operand ] OPERATOR … = [ outfile ]

Note: No space is allowed between the option flag and the associated arguments.

## Description

math will perform operations like add, subtract, multiply, and numerous other operands on one or more table data files or constants using Reverse Polish Notation (RPN) syntax. Arbitrarily complicated expressions may therefore be evaluated; the final result is written to an output file [or standard output]. Data operations are element-by-element, not matrix manipulations (except where noted). Some operators only require one operand (see below). If no data tables are used in the expression then options -T, -N can be set (and optionally -bo to indicate the data type for binary tables). If STDIN is given, the standard input will be read and placed on the stack as if a file with that content had been given on the command line. By default, all columns except the “time” column are operated on, but this can be changed (see -C). Complicated or frequently occurring expressions may be coded as a macro for future use or stored and recalled via named memory locations.

> Hewlett-Packard made lots of calculators (left) using Reverse Polish Notation, which is a post-fix system for mathematical notion originally developed by Jan_Łukasiewicz (right). Here, operands are entered first followed by an operator, e.g., “3 5 +” instead of “3 + 5 =” (photo courtesy of John W. Robbins).

## Required Arguments

operand

  If operand can be opened as a file it will be read as an ASCII (or binary, see -bi) table data file. If not a file, it is interpreted as a numerical constant or a special symbol (see below). The special argument STDIN means that standard input will be read and placed on the stack; STDIN can appear more than once if necessary.

outfile

  The name of a table data file that will hold the final result. If not given then the output is sent to standard output.

## Optional Arguments

-A t_f(t)[+e][+r][+s|w]

  Requires -N and will partially initialize a table with values from the given file t_f(t) containing t and f(t) only. The t is placed in column t_col while f(t) goes into column n_col - 1 (see -N) and is called b. The stack table is then the \(m \times (n+1)\) augmented matrix [ A | b ] and you want to solve for the least squares solution x to the matrix equation Ax = b. Usually, you will need to fill in the remaining columns in A using the various functions that defines the linear model you are trying to fit. If used with operators LSQFIT and SVDFIT you can optionally append some modifiers:

  - +e - Evaluate the solution and write a data set with four columns: t, f(t), the model solution and residuals at t, respectively [Default writes one column with model coefficients x].

  - +r - Only place f(t) (i.e., b) and leave the A part of the augmented matrix equation alone.

  - +s - Your t_f(t) has a third column with 1-sigma uncertainties, or

  - +w - Your t_f(t table has a third column with weights.

  Note: If either +s or +w are used we find the weighted solution. The weights (or sigmas) will be output as the last column if +e is in effect.

-C cols

  Select the columns that will be operated on until next occurrence of -C. List columns separated by commas; ranges like 1,3-5,7 are allowed, plus -Cx can be used for -C0 and -Cy can be used for -C1. -C (no arguments) resets the default action of using all columns except time column (see -N). -Ca selects all columns, including time column, while -Cr reverses (toggles) the current choices. When -C is in effect it also controls which columns from a file will be placed on the stack.

-E eigen

  Sets the minimum eigenvalue used by operators LSQFIT and SVDFIT [1e-7]. Smaller eigenvalues are set to zero and will not be considered in the solution.

-I

  Reverses the output row sequence from ascending time to descending [ascending].

-N n_col[/t_col]

  Select the number of columns and optionally the column number that contains the “time” variable [0]. Columns are numbered starting at 0 [2/0]. If input files are specified then -N will add any missing columns.

-Q[c|i|p|n]

  Quick mode for scalar calculation. Internally sets the equivalent of -Ca -N1/0 -T1. In this mode, constants may have dimensional units (i.e., c, i, or p), and will be converted to internal inches before computing. If one or more constants with units are encountered then the final answer will be reported in the unit set by PROJ_LENGTH_UNIT, unless overridden by appending another unit. Alternatively, append n for a non-dimensional result, meaning no unit conversion during output. To avoid any unit conversion on input, just do not use units.

-S[f|l]

  Only report the first or last row of the results [Default outputs all rows]. This is useful if you have computed a statistic (say the MODE) and only want to report a single number instead of numerous records with identical values. Append l to get the last row and f to get the first row only [Default].

-T[min/max/inc[+b|i|l|n]|file|list]

  Required when no input files are given. Builds an array for the “time” column (see -N). If there is no time column (i.e., your input has only data columns), give -T with no arguments; this also implies -Ca. For details on array creation, see Generate 1-D Array.

-V[level]

  Select verbosity level [w]. (See full description) (See cookbook information).

-bi record[+b|l] (more …)

  Select native binary format for primary table input.

-bo record[+b|l] (more …)

  Select native binary format for table output. [Default is same as input, but see -o]

-d[i|o][+c col]nodata (more …)

  Replace input columns that equal nodata with NaN and do the reverse on output.

-e[~]“pattern” | -e[~]/regexp/[i] (more …)

  Only accept data records that match the given pattern.

-f[i|o]colinfo (more …)

  Specify data types of input and/or output columns.

-gx|y|z|d|X|Y|D gap[u][+a][+c col][+n|p] (more …)

  Determine data gaps and line breaks.

-h[i|o][n][+c][+d][+m segheader][+r remark][+t title] (more …)

  Skip or produce header record(s).

-i cols[+l][+d divisor][+s scale|d|k][+o offset][,…][,t[word]] (more …)

  Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only).

-o cols[+l][+d divisor][+s scale|d|k][+o offset][,…][,t[word]] (more …)

  Select output columns and transformations (0 is first column, t is trailing text, append word to write one word only).

-q[i|o][~]rows|limits[+c col][+a|t|s] (more …)

  Select input or output rows or data limit(s) [all].

-s[cols][+a][+r] (more …)

  Set handling of NaN records for output.

-wy|a|w|d|h|m|s|c period[/phase][+c col] (more …)

  Convert an input coordinate to a cyclical coordinate.

-^ or just -

  Print a short message about the syntax of the command, then exit (Note: on Windows just use -).

-+ or just +

  Print an extensive usage (help) message, including the explanation of any module-specific option (but not the GMT common options), then exit.

-? or no arguments

  Print a complete usage (help) message, including the explanation of all options, then exit.

--PAR=value

  Temporarily override a GMT default setting; repeatable. See gmt.conf for parameters.

## Generate 1-D Array

We will demonstrate the use of options for creating 1-D arrays via gmtmath. Make an evenly spaced coordinate array from min to max in steps of inc, e.g.:

    gmt math -o0 -T3.1/4.2/0.1 T =
    3.1
    3.2
    3.3
    3.4
    3.5
    3.6
    3.7
    ...

Append +b if we should take \(\log_2\) of min and max, get their nearest integers, build an equidistant \(\log_2\)-array using inc integer increments in \(\log_2\), then undo the \(\log_2\) conversion. E.g., -T3/20/1+b will produce this sequence:

    gmt math -o0 -T3/20/1+b T =
    4
    8
    16

Append +l if we should take \(\log_{10}\) of min and max and build an array where inc can be 1 (every magnitude), 2, (1, 2, 5 times magnitude) or 3 (1-9 times magnitude). E.g., -T7/135/2+l will produce this sequence:

    gmt math -o0 -T7/135/2+l T =
    10
    20
    50
    100

For output values less frequently than every magnitude, use a negative integer inc:

    gmt math -o0 -T1e-4/1e4/-2+l T =
    0.0001
    0.01
    1
    100
    10000

Append +i if inc is a fractional number and it is cleaner to give its reciprocal value instead. To set up times for a 24-frames per second animation lasting 1 minute, run:

    gmt math -o0 -T0/60/24+i T =
    0
    0.0416666666667
    0.0833333333333
    0.125
    0.166666666667
    ...

Append +n if inc is meant to indicate the number of equidistant coordinates instead. To have exactly 5 equidistant values from 3.44 and 7.82, run:

    gmt math -o0 -T3.44/7.82/5+n T =
    3.44
    4.535
    5.63
    6.725
    7.82

Alternatively, let inc be a file with output coordinates in the first column, or provide a comma-separated list of specific coordinates, such as the first 6 Fibonacci numbers:

    gmt math -o0 -T0,1,1,2,3,5 T =
    0
    1
    1
    2
    3
    5

Notes: (1) If you need to pass the list nodes via a dataset file yet be understood as a list (i.e., no interpolation), then you must set the file header to contain the string “LIST”. (2) Should you need to ensure that the coordinates are unique and sorted (in case the file or list are not sorted or have duplicates) then supply the +u modifier.

If you only want a single value then you must append a comma to distinguish the list from the setting of an increment.

If the module allows you to set up an absolute time series, append a valid time unit from the list year, month, day, hour, minute, and second to the given increment; add +t to ensure time column (or use -f). Note: The internal time unit is still controlled independently by TIME_UNIT. The first 7 days of March 2020:

    gmt math -o0 -T2020-03-01T/2020-03-07T/1d T =
    2020-03-01T00:00:00
    2020-03-02T00:00:00
    2020-03-03T00:00:00
    2020-03-04T00:00:00
    2020-03-05T00:00:00
    2020-03-06T00:00:00
    2020-03-07T00:00:00

A few modules allow for +a which will paste the coordinate array to the output table.

Likewise, if the module allows you to set up a spatial distance series (with distances computed from the first two data columns), specify a new increment as inc with a geospatial distance unit from the list degree (arc), minute (arc), second (arc), meter, foot, kilometer, Miles (statute), nautical miles, or survey foot; see -j for calculation mode. To interpolate Cartesian distances instead, you must use the special unit c.

Finally, if you are only providing an increment and will obtain min and max from the data, then it is possible (max - min)/inc is not an integer, as required. If so, then inc will be adjusted to fit the range. Alternatively, append +e to keep inc exact and adjust max instead (keeping min fixed).

## Operators

Choose among the following operators. Here, “args” are the number of input and output arguments.

Operator | args | Returns | Type of Function
--- | --- | --- | ---
ABS | 1 1 | Absolute value of A | Arithmetic
ACOS | 1 1 | Inverse cosine (result in radians) | Calculus
ACOSD | 1 1 | Inverse cosine (result in degrees) | Calculus
ACOSH | 1 1 | Inverse hyperbolic cosine | Calculus
ACOT | 1 1 | Inverse of cotangent (result in radians) | Calculus
ACOTD | 1 1 | Inverse of cotangent (result in degrees) | Calculus
ACSC | 1 1 | Inverse of cosecant (result in radians) | Calculus
ACSCD | 1 1 | Inverse of cosecant (result in degrees) | Calculus
ADD | 2 1 | A + B (addition) | Arithmetic
AND | 2 1 | B if A equals NaN, else A | Logic
ASEC | 1 1 | Inverse of secant (result in radians) | Calculus
ASECD | 1 1 | Inverse of secant (result in degrees) | Calculus
ASIN | 1 1 | Inverse of sine (result in radians) | Calculus
ASIND | 1 1 | Inverse of sine (result in degrees) | Calculus
ASINH | 1 1 | Inverse of hyperbolic sine | Calculus
ATAN | 1 1 | Inverse of tangent (result in radians) | Calculus
ATAN2 | 2 1 | Inverse of tangent of A/B (result in radians) | Calculus
ATAN2D | 2 1 | Inverse of tangent of A/B (result in degrees) | Calculus
ATAND | 1 1 | Inverse of tangent (result in degrees) | Calculus
ATANH | 1 1 | Inverse of hyperbolic tangent | Calculus
BCDF | 3 1 | Binomial cumulative distribution function for p = A, n = B, and x = C | Probability
BEI | 1 1 | Kelvin function bei (A) | Special Functions
BER | 1 1 | Kelvin function ber (A) | Special Functions
BITAND | 2 1 | A & B (bitwise AND operator) | Logic
BITLEFT | 2 1 | A << B (bitwise left-shift operator) | Arithmetic
BITNOT | 1 1 | ~A (bitwise NOT operator, i.e., return two’s complement) | Logic
BITOR | 2 1 | A | B (bitwise OR operator) | Logic
BITRIGHT | 2 1 | A >> B (bitwise right-shift operator) | Arithmetic
BITTEST | 2 1 | 1 if bit B of A is set, else 0 (bitwise TEST operator) n | Logic
BITXOR | 2 1 | A ^ B (bitwise XOR operator) | Logic
BPDF | 3 1 | Binomial probability density function for p = A, n = B, and x = C | Probability
CEIL | 1 1 | ceil (A) (smallest integer >= A) | Logic
CHICDF | 2 1 | Chi-squared cumulative distribution function for chi2 = A and nu = B | Probability
CHICRIT | 2 1 | Chi-squared distribution critical value for alpha = A and nu = B | Probability
CHIPDF | 2 1 | Chi-squared probability density function for chi2 = A and nu = B | Probability
COL | 1 1 | Places column A on the stack | Special Operators
COMB | 2 1 | Combinations n_C_r, with n = A and r = B | Probability
CORRCOEFF | 2 1 | Correlation coefficient r(A, B) | Probability
COS | 1 1 | Cosine of A (A in radians) | Calculus
COSD | 1 1 | Cosine of A (A in degrees) | Calculus
COSH | 1 1 | Hyperbolic cosine of A | Calculus
COT | 1 1 | Cotangent of A (A in radians) | Calculus
COTD | 1 1 | Cotangent of A (A in degrees) | Calculus
CSC | 1 1 | Cosecant of A (A in radians) | Calculus
CSCD | 1 1 | Cosecant of A (A in degrees) | Calculus
D2DT2 | 1 1 | d2(A)/dt2 Central 2nd derivative | Calculus
D2R | 1 1 | Converts degrees to radians | Special Operators
DDT | 1 1 | d(A)/dt Central 1st derivative | Calculus
DEG2KM | 1 1 | Converts spherical degrees to kilometers | Special Operators
DENAN | 2 1 | Replace NaNs in A with values from B | Logic
DIFF | 1 1 | Forward difference between adjacent elements of A (A[1]-A[0], A[2]-A[1], …, NaN) | Arithmetic
DILOG | 1 1 | Dilogarithm (Spence’s) function | Special Functions
DIV | 2 1 | A / B (division) | Arithmetic
DUP | 1 2 | Places duplicate of A on the stack | Special Operators
ECDF | 2 1 | Exponential cumulative distribution function for x = A and lambda = B | Probability
ECRIT | 2 1 | Exponential distribution critical value for alpha = A and lambda = B | Probability
EPDF | 2 1 | Exponential probability density function for x = A and lambda = B | Probability
EQ | 2 1 | 1 if A equals B, else 0 | Logic
ERF | 1 1 | Error function erf (A) | Probability
ERFC | 1 1 | Complementary Error function erfc (A) | Probability
ERFINV | 1 1 | Inverse error function of A | Probability
EXCH | 2 2 | Exchanges A and B on the stack | Special Operators
EXP | 1 1 | E raised to a power. | Arithmetic
FACT | 1 1 | A! (A factorial) | Arithmetic
FCDF | 3 1 | F cumulative distribution function for F = A, nu1 = B, and nu2 = C | Probability
FCRIT | 3 1 | F distribution critical value for alpha = A, nu1 = B, and nu2 = C | Probability
FLIPUD | 1 1 | Reverse order of each column | Special Operators
FLOOR | 1 1 | greatest integer less than or equal to A | Logic
FMOD | 2 1 | A % B (remainder after truncated division) | Arithmetic
FPDF | 3 1 | F probability density function for F = A, nu1 = B, and nu2 = C | Probability
GE | 2 1 | 1 if A >= (greater or equal than) B, else 0 | Logic
GT | 2 1 | 1 if A > (greater than) B, else 0 | Logic
HSV2LAB | 3 3 | Convert h,s,v triplets to l,a,b triplets, with h = A (0-360), s = B and v = C (0-1) | Special Operators
HSV2RGB | 3 3 | Convert h,s,v triplets to r,g,b triplets, with h = A (0-360), s = B and v = C (0-1) | Special Operators
HSV2XYZ | 3 3 | Convert h,s,v triplets to x,t,z triplets, with h = A (0-360), s = B and v = C (0-1) | Special Operators
HYPOT | 2 1 | Hypotenuse of a right triangle of sides A and B (= sqrt (A2 + B2)) | Calculus
I0 | 1 1 | Modified Bessel function of A (1st kind, order 0) | Special Functions
I1 | 1 1 | Modified Bessel function of A (1st kind, order 1) | Special Functions
IFELSE | 3 1 | B if A is not equal to 0, else C | Logic
IN | 2 1 | Modified Bessel function of A (1st kind, order B) | Special Functions
INRANGE | 3 1 | 1 if B <= A <= C, else 0 | Logic
INT | 1 1 | Numerically integrate A | Calculus
INV | 1 1 | Invert (1/A) | Arithmetic
ISFINITE | 1 1 | 1 if A is finite, else 0 | Logic
ISNAN | 1 1 | 1 if A equals NaN, else 0 | Logic
J0 | 1 1 | Bessel function of A (1st kind, order 0) | Special Functions
J1 | 1 1 | Bessel function of A (1st kind, order 1) | Special Functions
JN | 2 1 | Bessel function of A (1st kind, order B) | Special Functions
K0 | 1 1 | Modified Kelvin function of A (2nd kind, order 0) | Special Functions
K1 | 1 1 | Modified Bessel function of A (2nd kind, order 1) | Special Functions
KEI | 1 1 | Kelvin function kei (A) | Special Functions
KER | 1 1 | Kelvin function ker (A) | Special Functions
KM2DEG | 1 1 | Converts kilometers to spherical degrees | Special Operators
KN | 2 1 | Modified Bessel function of A (2nd kind, order B) | Special Functions
KURT | 1 1 | Kurtosis of A | Probability
LAB2HSV | 3 3 | Convert l,a,b triplets to h,s,v triplets | Special Operators
LAB2RGB | 3 3 | Convert l,a,b triplets to r,g,b triplets | Special Operators
LAB2XYZ | 3 3 | Convert l,a,b triplets to x,y,z triplets | Special Operators
LCDF | 1 1 | Laplace cumulative distribution function for z = A | Probability
LCRIT | 1 1 | Laplace distribution critical value for alpha = A | Probability
LE | 2 1 | 1 if A <= (equal or smaller than) B, else 0 | Logic
LMSSCL | 1 1 | LMS (Least Median of Squares) scale estimate (LMS STD) of A | Probability
LMSSCLW | 2 1 | Weighted LMS scale estimate (LMS STD) of A for weights in B | Probability
LOG | 1 1 | log (A) (natural logarithm) | Arithmetic
LOG10 | 1 1 | log10 (A) (logarithm base 10) | Arithmetic
LOG1P | 1 1 | log (1+A) (natural logarithm, accurate for small A) | Arithmetic
LOG2 | 1 1 | log2 (A) (logarithm base 2) | Arithmetic
LOWER | 1 1 | The lowest (minimum) value of A | Arithmetic
LPDF | 1 1 | Laplace probability density function for z = A | Probability
LRAND | 2 1 | Laplace random noise with mean A and std. deviation B | Probability
LSQFIT | 1 0 | Stack is [A | b]; return least squares solution x = A \ b | Special Operators
LT | 2 1 | 1 if A < (smaller than) B, else 0 | Logic
MAD | 1 1 | Median Absolute Deviation (L1 STD) of A | Probability
MADW | 2 1 | Weighted Median Absolute Deviation (L1 STD) of A for weights in B | Probability
MAX | 2 1 | Maximum of A and B | Probability
MEAN | 1 1 | Mean value of A | Probability
MEANW | 2 1 | Weighted mean value of A for weights in B | Probability
MEDIAN | 1 1 | Median value of A | Probability
MEDIANW | 2 1 | Weighted median value of A for weights in B | Probability
MIN | 2 1 | Minimum of A and B | Probability
MOD | 2 1 | A mod B (remainder after floored division) | Arithmetic
MODE | 1 1 | Mode value (Least Median of Squares) of A | Probability
MODEW | 2 1 | Weighted mode value (Least Median of Squares) of A for weights in B | Probability
MUL | 2 1 | A x B (multiplication) | Arithmetic
NAN | 2 1 | NaN if A equals B, else A | Logic
NEG | 1 1 | Negative (-A) | Arithmetic
NEQ | 2 1 | 1 If A is not equal to B, else 0 | Logic
NORM | 1 1 | Normalize (A) so min(A) = 0 and max(A) = 1 | Probability
NOT | 1 1 | NaN ia A is equal NaN, 1 if A is equal to 0, else 0 | Logic
NRAND | 2 1 | Normal, random values with mean A and std. deviation B | Probability
OR | 2 1 | NaN if B equals NaN, else A | Logic
PCDF | 2 1 | Poisson cumulative distribution function for x = A and lambda = B | Probability
PERM | 2 1 | Permutations n_P_r, with n = A and r = B | Probability
PLM | 3 1 | Associated Legendre polynomial P(A) degree B order C | Special Functions
PLMg | 3 1 | Normalized associated Legendre polynomial P(A) degree B order C (geophysical convention) | Special Functions
POP | 1 0 | Delete top element from the stack | Special Operators
POW | 2 1 | A to the power of B | Arithmetic
PPDF | 2 1 | Poisson distribution P(x,lambda), with x = A and lambda = B | Probability
PQUANT | 2 1 | The B’th quantile (0-100%) of A | Probability
PQUANTW | 3 1 | The C’th weighted quantile (0-100%) of A for weights in B | Probability
PSI | 1 1 | Psi (or Digamma) of A | Special Functions
PV | 3 1 | Legendre function Pv(A) of degree v = real(B) + imag(C) | Special Functions
QV | 3 1 | Legendre function Qv(A) of degree v = real(B) + imag(C) | Special Functions
R2 | 2 1 | Hypotenuse squared (= A2 + B2) | Calculus
R2D | 1 1 | Convert radians to degrees | Special Operators
RAND | 2 1 | Uniform random values between A and B | Probability
RCDF | 1 1 | Rayleigh cumulative distribution function for z = A | Probability
RCRIT | 1 1 | Rayleigh distribution critical value for alpha = A | Probability
RGB2HSV | 3 3 | Convert r,g,b triplets to h,s,v triplets, with r = A, g = B, and b = C (in 0-255 range) | Special Operators
RGB2LAB | 3 3 | Convert r,g,b triplets to l,a,b triplets, with r = A, g = B, and b = C (in 0-255 range) | Special Operators
RGB2XYZ | 3 3 | Convert r,g,b triplets to x,y,z triplets, with r = A, g = B, and b = C (in 0-255 range) | Special Operators
RINT | 1 1 | Rint (A) (round to integral value nearest to A) | Arithmetic
RMS | 1 1 | Root-mean-square of A | Arithmetic
RMSW | 1 1 | Weighted root-mean-square of A for weights in B | Arithmetic
ROLL | 2 0 | Cyclically shifts the top A stack items by an amount B | Special Operators
ROOTS | 2 1 | Treats col A as f(t) = 0 and returns its roots | Special Operators
ROTT | 2 1 | Rotate A by the (constant) shift B in the t-direction | Arithmetic
RPDF | 1 1 | Rayleigh probability density function for z = A | Probability
SEC | 1 1 | Secant of A (A in radians) | Calculus
SECD | 1 1 | Secant of A (A in degrees) | Calculus
SIGN | 1 1 | Sign (+1 or -1) of A | Logic
SIN | 1 1 | Sine of A (A in radians) | Calculus
SINC | 1 1 | Normalized sinc function. | Special Functions
SIND | 1 1 | Sine of A (A in degrees) | Calculus
SINH | 1 1 | Hiperbolic sine of A | Calculus
SKEW | 1 1 | Skewness of A | Probability
SQR | 1 1 | Square (to the power of 2) | Arithmetic
SQRT | 1 1 | Square root | Arithmetic
STD | 1 1 | Standard deviation of A | Probability
STDW | 2 1 | Weighted standard deviation of A for weights in B | Probability
STEP | 1 1 | Heaviside step function H(A) | Special Functions
STEPT | 1 1 | Heaviside step function H(t-A) | Special Functions
SUB | 2 1 | A - B (subtraction) | Arithmetic
SUM | 1 1 | Cumulative sum of A | Arithmetic
SVDFIT | 1 0 | Stack is [A | b]; return x = A \ b via SVD decomposition (see |-E)| | Special Operators
TAN | 1 1 | Tangent of A (A in radians) | Calculus
TAND | 1 1 | Tangent of A (A in degrees) | Calculus
TANH | 1 1 | Hyperbolic tangent of A | Calculus
TAPER | 1 1 | Unit weights cosine-tapered to zero within A of end margins | Special Operators
TCDF | 2 1 | Student’s t cumulative distribution function for t = A, and nu = B | Probability
TCRIT | 2 1 | Student’s t distribution critical value for alpha = A and nu = B | Probability
TN | 2 1 | Chebyshev polynomial Tn(-1<A<+1) of degree B | Special Functions
TPDF | 2 1 | Student’s t probability density function for t = A, and nu = B | Probability
UPPER | 1 1 | The highest (maximum) value of A | Arithmetic
VAR | 1 1 | Variance of A | Probability
VARW | 2 1 | Weighted variance of A for weights in B | Probability
VPDF | 3 1 | Von Mises density distribution V(x,mu,kappa), with angles = A, mu = B, and kappa = C | Probability
WCDF | 3 1 | Weibull cumulative distribution function for x = A, scale = B, and shape = C | Probability
WCRIT | 3 1 | Weibull distribution critical value for alpha = A, scale = B, and shape = C | Probability
WPDF | 3 1 | Weibull density distribution P(x,scale,shape), with x = A, scale = B, and shape = C | Probability
XOR | 2 1 | B if A equals NaN, else A | Logic
XYZ2HSV | 3 3 | Convert x,y,z triplets to h,s,v triplets | Special Operators
XYZ2LAB | 3 3 | Convert x,y,z triplets to l,a,b triplets | Special Operators
XYZ2RGB | 3 3 | Convert x,y,z triplets to r,g,b triplets | Special Operators
Y0 | 1 1 | Bessel function of A (2nd kind, order 0) | Special Functions
Y1 | 1 1 | Bessel function of A (2nd kind, order 1) | Special Functions
YN | 2 1 | Bessel function of A (2nd kind, order B) | Special Functions
ZCDF | 1 1 | Normal cumulative distribution function for z = A | Probability
ZCRIT | 1 1 | Normal distribution critical value for alpha = A | Probability
ZPDF | 1 1 | Normal probability density function for z = A | Probability

## Symbols

The following symbols have special meaning:

PI | 3.1415926…
E | 2.7182818…
EULER | 0.5772156…
PHI | 1.6180339… (golden ratio)
EPS_F | 1.192092896e-07 (sgl. prec. eps)
EPS_D | 2.2204460492503131e-16 (dbl. prec. eps)
TMIN | Minimum t value
TMAX | Maximum t value
TRANGE | Range of t values
TINC | t increment
N | The number of records
T | Table with t-coordinates
TNORM | Table with normalized t-coordinates
TROW | Table with row numbers 1, 2, …, N-1

## ASCII Format Precision

The ASCII output formats of numerical data are controlled by parameters in your gmt.conf file. Longitude and latitude are formatted according to FORMAT_GEO_OUT, absolute time is under the control of FORMAT_DATE_OUT and FORMAT_CLOCK_OUT, whereas general floating point values are formatted according to FORMAT_FLOAT_OUT. Be aware that the format in effect can lead to loss of precision in ASCII output, which can lead to various problems downstream. If you find the output is not written with enough precision, consider switching to binary output (-bo if available) or specify more decimals using the FORMAT_FLOAT_OUT setting.

## Notes On Operators

1. The operators PLM and PLMg calculate the associated Legendre polynomial of degree L and order M in x which must satisfy \(-1 \leq x \leq +1\) and \(0 \leq M \leq L\). Here, x, L, and M are the three arguments preceding the operator. PLM is not normalized and includes the Condon-Shortley phase \((-1)^M\). PLMg is normalized in the way that is most commonly used in geophysics. The Condon-Shortley phase can be added by using -M as argument. PLM will overflow at higher degrees, whereas PLMg is stable until ultra high degrees (at least 3000).

2. Files that have the same names as some operators, e.g., ADD, SIGN, =, etc. should be identified by prepending the current directory (i.e., ./).

3. The stack depth limit is hard-wired to 100.

4. All functions expecting a positive radius (e.g., LOG, KEI, etc.) are passed the absolute value of their argument.

5. The DDT and D2DT2 functions only work on regularly spaced data.

6. All derivatives are based on central finite differences, with natural boundary conditions.

7. ROOTS must be the last operator on the stack, only followed by =.

## STORE, RECALL and CLEAR

You may store intermediate calculations to a named variable that you may recall and place on the stack at a later time. This is useful if you need access to a computed quantity many times in your expression as it will shorten the overall expression and improve readability. To save a result you use the special operator STO@label, where label is the name you choose to give the quantity. To recall the stored result to the stack at a later time, use [RCL]@label, i.e., RCL is optional. To clear memory you may use CLR@label. Note that both STO and CLR leave the stack unchanged.

1. The bitwise operators (BITAND, BITLEFT, BITNOT, BITOR, BITRIGHT, BITTEST, and BITXOR) convert a tables’s double precision values to unsigned 64-bit ints to perform the bitwise operations. Consequently, the largest whole integer value that can be stored in a double precision value is 253 or 9,007,199,254,740,992. Any higher result will be masked to fit in the lower 54 bits. Thus, bit operations are effectively limited to 54 bits. All bitwise operators return NaN if given NaN arguments or bit-settings <= 0.

2. TAPER will interpret its argument to be a width in the same units as the time-axis, but if no time is provided (i.e., plain data tables) then the width is taken to be given in number of rows.

3. The color-triplet conversion functions (RGB2HSV, etc.) includes not only r,g,b and h,s,v triplet conversions, but also l,a,b (CIE L a b ) and sRGB (x,y,z) conversions between all four color spaces. These functions behave differently whether -Q is used or not. With -Q we expect three input constants and we place three output results on the stack. Since only the top stack item is printed, you must use operators such as POP and ROLL to get to the item of interest. Without -Q, these operators work across the three columns and modify the three column entries, returning their result as a single three-column item on the stack.

4. The VPDF operator expects angles in degrees.

## Macros

Users may save their favorite operator combinations as macros via the file gmtmath.macros in their current or user directory. The file may contain any number of macros (one per record); comment lines starting with # are skipped. The format for the macros is name = arg1 arg2 … arg2 [ : comment] where name is how the macro will be used. When this operator appears on the command line we simply replace it with the listed argument list. No macro may call another macro. As an example, the following macro expects that the time-column contains seafloor ages in Myr and computes the predicted half-space bathymetry:

DEPTH = SQRT 350 MUL 2500 ADD NEG : usage: DEPTH to return half-space seafloor depths

Note: Because geographic or time constants may be present in a macro, it is required that the optional comment flag (:) must be followed by a space. As another example, we show a macro GPSWEEK which determines which GPS week a timestamp belongs to:

GPSWEEK = 1980-01-06T00:00:00 SUB 86400 DIV 7 DIV FLOOR : usage: GPS week without rollover

## Active Column Selection

When -C cols is set then any operation, including loading of data from files, will restrict which columns are affected. To avoid unexpected results, note that if you issue a -C cols option before you load in the data then only those columns will be updated, hence the unspecified columns will be zero. On the other hand, if you load the file first and then issue -C cols then the unspecified columns will have been loaded but are then ignored until you undo the effect of -C.

## Absolute Time Column(s)

If input data have more than one column and the “time” column (set via -N [0]) contains absolute time, then the default output format for any other columns containing absolute time will be reset to relative time. Likewise, in scalar mode (-Q) the time column will be operated on and hence it also will be formatted as relative time. Finally, if -C is used to include “time” in the columns operated on then we likewise will reset that column’s format to relative time. The user can override this behavior with a suitable -f or -fo setting. Note: We cannot guess what your operations on the time column will do, hence this default behavior. As examples, if you are computing time differences then clearly relative time formatting is required, while if you are computing new absolute times by, say, adding an interval to absolute times then you will need to use -fo to set the output format for such columns to absolute time.

## Scalar Math with Units

If you use -Q to do simple calculations, please note that the support for dimensional units is limited to converting a number ending in c, i, or p to internal inches. Thus, while you can run “gmt -Qc 1c 1c MUL =”, you may be surprised that the output area is not 1 cm squared. The reason is that gmt math cannot keep track of what unit any particular item on the stack might be so it will assume it is internally in inches and then scale the final output to cm. In this particular case, the unit is in inches squared and scaling by 2.54 once will give 0.3937 inch times cm as the unit. Thus, conversions only work for linear unit calculations, such as gmt math -Qp 1c 0.5i ADD =, which will return the result as 64.34 points.

## Trailing Text

Any trailing text in the first input file will be passed to the output data set. You can turn off the output of text with -on.

## Examples

Note: Below are some examples of valid syntax for this module. The examples that use remote files (file names starting with `@`) can be cut and pasted into your terminal for testing. Other commands requiring input files are just dummy examples of the types of uses that are common but cannot be run verbatim as written.

To add two plot dimensions of different units, we can run

    length=$(gmt math -Q 15c 2i SUB =)

To compute the ratio of two plot dimensions of different units, we select non-dimensional output and run

    ratio=$(gmt math -Qn 15c 2i DIV =)

To take the square root of the content of the second data column being piped through gmtmath by process1 and pipe it through a 3rd process, use

    process1 | gmt math STDIN SQRT = | process3

To take \(\log_{10}\) of the average of 2 data files, use

    gmt math file1.txt file2.txt ADD 0.5 MUL LOG10 = file3.txt

Given the file samples.txt, which holds seafloor ages in m.y. and seafloor depth in m, use the relation depth(in m) = 2500 + 350 * sqrt (age) to print the depth anomalies:

    gmt math samples.txt T SQRT 350 MUL 2500 ADD SUB = | lpr

To take the average of columns 1 and 4-6 in the three data sets sizes.1, sizes.2, and sizes.3, use

    gmt math -C1,4-6 sizes.1 sizes.2 ADD sizes.3 ADD 3 DIV = ave.txt

To take the 1-column data set ages.txt and calculate the modal value and assign it to a variable, try

    mode_age=$(gmt math -S -T ages.txt MODE =)

To evaluate the dilog(x) function for coordinates given in the file t.txt:

    gmt math -Tt.txt T DILOG = dilog.txt

To demonstrate the use of stored variables, consider this sum of the first 3 cosine harmonics where we store and repeatedly recall the trigonometric argument (2*pi*T/360):

    gmt math -T0/360/1 2 PI MUL 360 DIV T MUL STO@kT COS @kT 2 MUL COS ADD @kT 3 MUL COS ADD = harmonics.txt

To use gmtmath as a RPN Hewlett-Packard calculator on scalars (i.e., no input files) and calculate arbitrary expressions, use the -Q option. As an example, we will calculate the value of Kei (((1 + 1.75)/2.2) + cos (60)) and store the result in the shell variable z:

    z=$(gmt math -Q 1 1.75 ADD 2.2 DIV 60 COSD ADD KEI =)

To convert the r,g,b value for yellow to h,s,v and save the hue, try

    hue=$(gmt math -Q 255 255 0 RGB2HSV POP POP =)

To use gmtmath as a general least squares equation solver, imagine that the current table is the augmented matrix [ A | b ] and you want the least squares solution x to the matrix equation A * x = b. The operator LSQFIT does this; it is your job to populate the matrix correctly first. The -A option will facilitate this. Suppose you have a 2-column file ty.txt with t and y and you would like to fit a the model y(t) = a + b*t + c*H(t-t0), where H(t) is the Heaviside step function for a given t0 = 1.55. Then, you need a 4-column augmented table loaded with t in column 1 and your observed y in column 3. The calculation becomes

    gmt math -N4/1 -Aty.txt -C0 1 ADD -C2 1.55 STEPT ADD -Ca LSQFIT = solution.txt

Note we use the -C option to select which columns we are working on, then make active all the columns we need (here all of them, with -Ca) before calling LSQFIT. The second and fourth columns (col numbers 1 and 3) are preloaded with t and y, respectively, the other columns are zero. If you already have a pre-calculated table with the augmented matrix [ A | b ] in a file (say lsqsys.txt), the least squares solution is simply

    gmt math -T lsqsys.txt LSQFIT = solution.txt

Users must be aware that when -C controls which columns are to be active the control extends to placing columns from files as well. Contrast the different result obtained by these very similar commands:

    echo 1 2 3 4 | gmt math STDIN -C3 1 ADD =
    1    2    3    5

versus

    echo 1 2 3 4 | gmt math -C3 STDIN 1 ADD =
    0    0    0    5

To calculate how many days there were in the 80s decade:

    gmt math -Q 1991-01-01T 1981-01-01T SUB --TIME_UNIT=d =

To determine the 1000th day since the beginning of the millennium:

    gmt math -Q 2001-01-01T 1000 ADD --TIME_UNIT=d -fT =

## References

Abramowitz, M., and I. A. Stegun, 1964, Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, Dover, New York.

Holmes, S. A., and W. E. Featherstone, 2002, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalized associated Legendre functions. Journal of Geodesy, 76, 279-299.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992, Numerical Recipes, 2nd edition, Cambridge Univ., New York.

Spanier, J., and K. B. Oldman, 1987, An Atlas of Functions, Hemisphere Publishing Corp.

More on Reverse Polish Notation

Reverse Polish Notation (RPN) or postfix notation is widely used in computer science because of the simplicity of implementing a stack-based computation system. The PostScipt language we used to make GMT graphics is postfix, for instance. To learn more about RPN or postfix, watch this YouTube video explanation:

## See Also

gmt, grdmath