SPECPR Users’ Manual Page 8.1

CHAPTER 8

MATH OPERATIONS

8.1 Introduction All mathematical operations are carried out in Math Operations except for extinction analysis. The math routines also contain the "special functions." Special functions do some operation on a data set, or generate a data set. While most special functions are math related, they need not be (for example sort a data set into increasing wavelengths). The general sequence the user sees in the Math Operations section is the following:

• At the math main menu, the user enters a math command • The math function displays what it is going to do and the user works his/her way through the function-specific menus. • When the math function completes normally, the user is asked where the data should be written (which v, w, d, u, or y data file and what record number). The data are not written at this time; that is the second to last step below. The ultimate location is requested at this point so that the spectrum is properly identified in case print-outs are made. • The user is asked to type in a title to the data set. • The data are plotted in the graphics terminal (with the crt plot routines, see Chapter 9). In the crt plot routines, the user can change or add to the header information. • The data are written to the data file when the user soft exits from the crt plot routines (exit with e). See warnings and important rules below. • The user is returned to the math operations menu. |

WARNING: the number of data channels operated on by any math routine is determined by the number of channels in the data set in use (displayed in the top few lines of each routine). It is UP TO THE USER to be sure the number of channels is set properly. Improper number of channels is a very common reason for improper results.

WARNING: the number of channels in the data set written by the math routines is controlled by the number of channels in use AT THE TIME the user EXITS the CRT PLOT ROUTINES. If you change the number of channels in the header routines, it will have no effect--that value is overridden by the math writing routine.

IMPORTANT RULE: the wavelength pointer will be set automatically by the math routine at the time the data are written to the data file IF the wavelength set in use IS IN THE SAME DATA FILE.

For example, if you divide two spectra in data file "w", and write the results in data file "v", use a wavelength set that is already in "v". Thus if: |

w 10 = wavelengths with 56 channels and you want to divide w11 by w12, first put a copy of w10 into the "v" file (see Chapter 10 on how to copy/transfer), then assign the "v" wavelength set before you do the division. The wavelength pointer will then be set automatically when the results from the math operation are written to the v-file. |

8.1.1 Multiple commands from the math command menu. One entire line (80 characters) of commands may be typed in for execution. Each math command must be separated by a comma.

Example: w11/w12,v23*c5.62e,y236/w12 |

After each math operation is completed, the next command is executed without returning to the math menu.

8.1.2 Menu Information The information in the Math menu can be turned off by typing

in |

or turned back on by typing

i |

and nothing else on the command line.

8.2 Subtraction, Multiplication, and Division Generally these operations contain a file ID (v, w, u, d, or y), record number, operation sign

- for subtraction, * for multiplication, and / for division,

then the second file ID (v, w, d, c, u, y, or s; s is used in division only), the second record number, and the options. Options include:

e to include 1 sigma error bars, n to do a subtraction without airmass calculation, b for turn on band normalization (section 8.9), bn for turn off band normalization, t for change sidereal time in subtraction with airmass calculation, r for change RA and declination in subtraction calculation, and p for production processing. |

The production processing option stops only to request the title and writes the result to device v without doing the CRT plot. This option (p) makes processing equivalent to the time spent in performing batch processing. When you enter the output title (section 8.8), you may turn off (or on) production processing by typing "pn" (or "po" for on) after the title. See section 8.8. The c in place of the second file ID is for constant. The value of the constant is then placed where the second record number is located.

Example: |

w10-11, w12-13, v10*c1.3, v23/46eb, v12-d10np bn Here file w record 11 is subtracted from 10 (the default when the second file ID is left out is the first file ID). Next w13 is subtracted from w12, then v10 is multiplied by the constant 1.3, next v23 is divided by v46 with 1 sigma standard deviation (errors) included and also a band normalization performed. On the math command line, spaces may be included at any point except within a number, or they may be completely left out. The program processes the commands sequentially. When performing each operation, the program enters the appropriate routine, writes the titles of the requested files on the CRT, and pauses for a continue command (unless the Production option has been set). If you type x |

here, the program exits directly to math operations. If you type

e |

the program exits that operation and begins processing of the next operation if one exists. The production option is set to no production at the end of each operation whereas the band normalization option stays on until turned off by a

bn |

command. When the operation is completed, the program goes to section 8.1. Math Operations can be used to change the wavelength file set in use. Example:

v23/v45V2, v86*v27W46 |

where the capital file ID specifies a wavelength set within the corresponding data file. The number of channels in use will then be read from the wavelength set.

Special Note on the Division Routine Division by a small number (-10-36 to +10-36) is defined as a deleted point

(-1.23x10+34 = A DELETED POINT) |

x by this program.

Special Notes on the Subtraction Routine Normal subtraction calculates airmass based on the coordinates found in the first 13 characters of the title of the first input file; if these coordinates are not entered in the correct format (HHMMSS+DDMMSS no spaces, declination sign must be explicitly specified as plus or minus, and any one of the first six digits must be non-zero), the routine issues the command

TYPE IN RIGHT ASCENSION AND DECLINATION |

and awaits a correct response. The routine repeats this message until a correct response is encountered--e and x are NOT correct responses. The correct response is up to six numbers separated by spaces, any one of the first three of which must be non-zero, the fourth of which should be signed only if negative, and all the rest of which should be unsigned.

TO GET OUT OF THIS LOOP, enter any single non-zero number, and type e or x at the next possible opportunity.

(This format may be confusing because the interpretation routine from the first 13 characters requires an explicitly specified plus or minus sign for the declination, and the request for RA and DEC cannot interpret plus signs.)

8.3 Addition Routine The addition routine can add from 2 to 128 spectra with deletion of zeros or requested points. To access the addition routine, type + and any options:

+ no options +e addition with propagation of existing error bars +s sum +a average +se sum with errors +ae average with errors |

with no other math operations after that on the input line (because they will not be processed due to a programming restriction). Upon entering the addition routine, you will be asked whether you want to average or sum (unless you specified the a or s option as given above). Type

a to Average, s to Sum and, Next type: z to DELETE ZEROS in the input data, v to DELETE data values outside a user set limit, and/or s to DELETE data values outside a certain standard deviation limit. Next you will be given instructions on entering the files to be added or summed. Type in the file ID [v, w, d, u, y or c (for constant)] and the record number or constant. If you wish to delete points from this record, type d after the record number or constant. If you elect to delete channels, you may start the channel number list after the "d". You may use more than one line to input these numbers, but each channel to be deleted from the sum or average must be listed. When you are through entering numbers to be deleted, type a "c" to continue. Example: v23 \# normal input, no deletions, v23 d 1 2 26t32 c \# delete channels 1, 2, 26 through 32. |

When you are through entering record numbers, type

b |

to begin analysis. Note only 1 constant can be entered per addition run. If you type

e |

or

x |

the program will exit directly back to math operations; otherwise, after the analysis is complete, the program goes to section 8.7.

8.4 Error Analysis Error analyses are included in addition, multiplication and division, and subtraction and many of the special functions (see individual functions - section 8.6) In all cases of error analysis, the data numbers should be within the range of

1.0×10-15 to 1.0×10+15 |

to avoid overflow. If overflow occurs, the error is set to zero. If the data is zero in the multiplication or division routines, when using the propagation of errors, the error is set to zero for that channel. The errors represent 1 sigma standard deviation of the mean and are normally first computed in the addition routine as a standard deviation of the mean computation. The equations used in propagating errors are as follows: .KE Given a± σa, b ± σb , c ± σc and the result x ± σx the equations used are:

*division:* HTML-IMAGEcpile { X ~=~ a over b above
sigma sub X ~=~ left [ left ( sigma sub a over a right ) sup
2 + left ( sigma sub b over b right ) sup 2 right ] sup {1
over 2} X } HTML-IMAGE-END.KE

*multiplication:* HTML-IMAGEcpile { X ~=~ a b above
sigma sub X ~=~ left [ left ( sigma sub a over a right ) sup
2 + left ( sigma sub b over b right ) sup 2 right ] sup {1
over 2} X } HTML-IMAGE-END.KE

*subtraction:* HTML-IMAGEcpile { X ~=~ a - b above
sigma sub X ~=~ left { sigma sub a sup 2 + sigma sub b sup 2
right } sup { 1 over 2} } HTML-IMAGE-END.KE

*addition* (when errors already exist):
HTML-IMAGE(sum):~~ X ~=~ a + b + . . . +c, HTML-IMAGE-END.EQ
sigma sub x ~=~ left { sigma sub a sup 2 + sigma sub b sup 2
+ . . HTML-IMAGE-END

HTML-IMAGE(average~ of~ n~ spectra):~~ X ~=~ (a + b + . . . c) / n above HTML-IMAGE-END.EQ sigma sub x ~=~ left { sigma sub a sup 2 + sigma sub b sup 2 + . . HTML-IMAGE-END.KE

standard deviation of the mean is derived from (generate errors for the first time):

╭ n _ ╮½ │ Σ (xi - x)2 │ σx = │ i=1 │ │ ————— │ ╰ (n - 1) ╯

where n = number of spectra, xi = each spectrum, and HTML-IMAGEx bar ~=~ 1 over n sum from i=1 to n x sub i HTML-IMAGE-END.KE

8.5 Algebraic and Trigonometric Functions

A complete set of algebraic, trigonometric, and power functions exists similar to those on a scientific calculator. No error analysis is included with any of these functions. The functions are accessed by file ID, file number, colon (:), and the name of the function. Functions are:

Exponential: exp Natural logarithm: ln Common logarithm: log Ten to spectrum power: 10**x Inverse: 1/x Spectrum to the constant power: x**c Constant to the spectrum power: c**x Sine (in radians): sin Cosine (in radians): cos Tangent (in radians): tan Sine (in degrees): sind Cosine (in degrees): cosd Tangent (in degrees): tand Inverse sine (radians): invsin Inverse cosine (radians): invcos Inverse tangent (radians): invtan Inverse sine (degrees): invsind Inverse cosine (degrees): invcosd Inverse tangent (degrees): invtand Hyperbolic sine (radians): sinh Hyperbolic cosine (radians): cosh Hyperbolic tangent (radians): tanh Absolute value: abs Integer value: int Fractional part: frac Examples: |

v397:ln \# natural log of v397 w76:cos \# cosine of w76. |

In both the spectrum to the constant power and constant to the spectrum power, the value of the constant is after the command (x**c or c**x):

v397:x**c3.65 \# v397 to the 3.65 power, v397:c**x3.65 \# constant 3.65 to the v397 power |

No options are valid with any of these operations including errors (except 1/x can include errors), band normalization, and production processing. This is due to programming limits (there has to be some). Since these routines are not used too often, this should make little difference. What are they used for? Normal spectral processing does not need them, but say you wanted to make a plot of a phase function for a Lambert sphere:

HTML-IMAGEPHI ( alpha ) ~=~ 1 over pi left [ sin~ alpha + ( pi - alpha ) cos~ alpha right ] HTML-IMAGE-END You could generate data for α increasing from 0 to 180° using the wavelength routines and then this function using the math functions. The result can then be plotted on the Gould printer plotter using the plot routines.

8.6 Special Functions Special function subroutines are additional routines for operating on any kind of data. They do any operation: for example a data list, a data editor, or a specific computation. Some special functions operate on a data set (for example, a smoothing function) while others may create a new data set (for example, read one from an ascii text file or digitize one on a digitizing tablet, or simply compute one, like a Plank black body). Special functions are called from math operations by one of two methods, depending on whether or not the function operates on an existing data set. If a function requires a data set, you enter the file id, record number, an "f" followed by the function number. For example, to smooth (function 7) data set v23, you would enter the command:

v23f7 |

Special functions also have alias names. For example, the smoothing function f7 has the alias "smooth". Thus

v23f7 |

and

v23smooth |

are equivalent. Some special functions do not need an input data set. In that case, just specify the function or its alias. For example, to compute a Planck black body, type:

f6 |

or

planck |

and the function will be started. Note there is no colon in the command line for special functions as compared to the algebraic and trig routines. Special function 1 is a function which lists a catalog of all available special functions. Just type "f1" from the math operations command line. This list contains a code at the end of the short description which tells if the function needs a data file input before the function on the command line. If this is the case, you will see an "(f)" at the end of the description. If the function can also propagate the error bars to the data set, it will have an "(f,e)" at the end of the description. To propagate error bars for those routines that support error propagation, you must specify an "e" option on the command line. For example, to propagate error bars in the cubic spline interpolation routine, function f12, alias cspline, and operate on data set v23, type the command:

v23f12e |

or

v23cspline e |

Specpr is designed to add special functions easily. Subroutine calls already exist in the program for many as yet unwritten routines. These routines are designed for easy linking to the present program. New routines can be user written with less effort than a completely separate program since, in most cases, SPECPR handles all the data management.

8.7 Return from a Math Operation or Function When a routine such as addition, subtraction, special function, trig. function, etc., has finished, the program asks the user where he/she wishes the data to be written. The data can be written to any of the files: v, w, d, u, or y if allowed by the protection. If the protection on the device is positive or zero, the data can only be written to the protection value plus 1. Thus, it is not necessary to type in this value since it is the only value possible. Simply type the file letter ID and return; the record number will be set automatically. The data set is not written at this point. It is written after the CRT plot routines. The user may also exit the current operation from this point. By typing

e |

the current operation is terminated and the program begins execution of the next command if there is one or returns to Math Operations if there are no other command requests. If the user types

x |

all processing stops, and the program returns directly to Math Operations. If the user requests the data to be stored, then the program continues to the titles routine (section 8.8).

8.8 The Titles Routine The titles routine displays 2 titles on the CRT which may be selected for the new data, or one of 25 user stored titles can be recalled, or the user can type in a new title and store it. The 2 titles displayed are as follows.

The "option p title" This is the "present title" of the first data set read in for 2 file operations or the title of the last data set in the addition routine or many other routines. Some special functions may generate a suggested present title.

The "option l title" This is the title requested for the last operation.

In the subtraction routine, the program decodes an "object-sky" title as the option p title. This is based on inputs from the "Wedge" CVF spectrometer data system and may not be valid with other types of subtraction. All titles are 40 characters long, so to select the option p or l titles, type

p |

or

l |

with no other characters in columns to 40. To recall one of the stored titles, type

t# |

where the "#" is the title number (1 to 25). There can be no other characters in columns 1 to 40. If other characters are detected here, the program will think this is a new title and use it. The title used, whether recalled or new, is stored in title file t1. To list the contents of the title file, type

tl |

with no other characters on the command line. The titles will be listed and the program will again ask for a title. Once a title has been selected (recalled or new), further commands can be placed in columns 40 or greater. To store a title, type

t# |

where the "#" is the title number (3 to 25 are valid). Remember, the "t#" must be positioned in columns 41 or greater with the requested title (recalled or new) in columns 1 through 40. The Band normalization or Production processing options can also be turned on or off by control characters in columns 41 or greater. Type

b |

to turn on or

bn |

to turn off Band normalization. Type

p |

to turn on or

pn |

to turn off the Production processing option (the n is for none). The vertical scale of the pending crt plot can also be changed after column 41 of the title command line. Simply type "c and the lower and upper range. For example, to change the scale from 0.92 to 1.63, type

c 0.92 1.63 |

after column 41. An example of a complete title entry would be

Scapolite HS351 / Halon 4x resol .2-3um t5 b c .5 1.5 |

which means to make the title of the current data set "Scapolite HS351 / Halon 4x resol .2-3um", store it in title location 5, turn on band normalization and change scale of the pending CRT plot to lower bound 0.5 and upper bound 1.5. The current operation can be terminated as in section 8.7 by typing

e |

or

x |

with no other characters on the line. Typing "e" will terminate the present operation, and the program will begin the next operation if there is one. Otherwise, it will return to Math Operations. Typing "x" will terminate all operations and will go directly to Math Operations. If a title is requested, the program will then go to the Band Normalization Routine (section 8.9) if the Band Normalization option is on or to the CRT plot routines if the Band Normalization and Production options are off. If the Production and Band Normalization options are on, the program will go to the Band Normalization; otherwise, if only the Production option is on it will write the data and begin processing the next command.

8.9 Band Normalization The Band Normalization is a least squares analysis over a user selected band with the spectrum scaled so the middle of the band is scaled to unity. The band can be one channel or however many channels are in the spectrum. The Band Normalization routine is called after each operation if the Band Normalization option is on. When you are in the band normalization routine, you have several options to control the region that will be normalized. Channels can be deleted inside the region of normalization by typing the

d |

command. The program then asks for the points. One line can be filled with points to be deleted (numbers, no characters). The band limits can be moved by typing the

m |

command. Deleted channels can be reinserted by typing the

r |

command. The band area with the fitted line along with the correlation coefficient, the previous Band Normalization factor (the normalization factor of the data before this normalization), the current normalization factor (as determined by this normalization), and the future normalization factor (if this normalization is carried out) are plotted on the CRT. The future normalization factor equals the previous times the current factors. WARNING: if the previous normalization factor was zero when the Band Normalization routine is entered, it is reset to 1.0. To perform the normalization of the data, type

b |

otherwise, type

e |

to exit the routine without changing the normalization factor. The "e" soft exit continues to the next step which will normally be the CRT plot routine unless production processing is turned on. If production processing is turned on, the specpr will write the data when exiting band normalization with an "e" soft exit. A hard exit, command "x" from the Band Normalization routine, will terminate all pending commands and return to the Math Operations command line. Default Band Normalization limits when program is begun are 30 to 38.