fortran 90 code chapters b1–b20 - pudn.comread.pudn.com/downloads174/ebook/810137/numerical...

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Fortran 90 Code Chapters B1–B20

Fortran 90 versions of all the Numerical Recipes routines appear in the followingChapters B1 through B20, numbered in correspondence with Chapters 1 through 20in Volume 1. Within each chapter, the routines appear in the same order as in Volume1, but not broken out separately by section number within Volume 1’s chapters.

There are commentaries accompanying many of the routines, generally follow-ing the printed listing of the routine to which they apply. These are of two kinds:issues related to parallelizing the algorithm in question, and issues related to theFortran 90 implementation. To distinguish between these two, rather different, kindsof discussions, we use the two icons,

f90the left icon (above) indicating a “parallel note,” and the right icon denoting a“Fortran 90 tip.” Specific code segments of the routine that are discussed in thesecommentaries are singled out by reproducing some of the code as an “index line”next to the icon, or at the beginning of subsequent paragraphs if there are severalitems that are commented on.

d=merge(FPMIN,d,abs(d)

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Chapter B1. Preliminaries

SUBROUTINE flmoon(n,nph,jd,frac)USE nrtype; USE nrutil, ONLY : nrerrorIMPLICIT NONEINTEGER(I4B), INTENT(IN) :: n,nphINTEGER(I4B), INTENT(OUT) :: jdREAL(SP), INTENT(OUT) :: frac

Our programs begin with an introductory comment summarizing their purpose and explain-ing their calling sequence. This routine calculates the phases of the moon. Given an integern and a code nph for the phase desired (nph = 0 for new moon, 1 for first quarter, 2 forfull, 3 for last quarter), the routine returns the Julian Day Number jd, and the fractionalpart of a day frac to be added to it, of the nth such phase since January, 1900. GreenwichMean Time is assumed.

REAL(SP), PARAMETER :: RAD=PI/180.0_spINTEGER(I4B) :: iREAL(SP) :: am,as,c,t,t2,xtrac=n+nph/4.0_sp This is how we comment an individual line.t=c/1236.85_spt2=t**2as=359.2242_sp+29.105356_sp*c You aren’t really intended to understand this al-

gorithm, but it does work!am=306.0253_sp+385.816918_sp*c+0.010730_sp*t2jd=2415020+28*n+7*nphxtra=0.75933_sp+1.53058868_sp*c+(1.178e-4_sp-1.55e-7_sp*t)*t2select case(nph)

case(0,2)xtra=xtra+(0.1734_sp-3.93e-4_sp*t)*sin(RAD*as)-0.4068_sp*sin(RAD*am)

case(1,3)xtra=xtra+(0.1721_sp-4.0e-4_sp*t)*sin(RAD*as)-0.6280_sp*sin(RAD*am)

case defaultcall nrerror(’flmoon: nph is unknown’) This is how we will indicate error

conditions.end selecti=int(merge(xtra,xtra-1.0_sp, xtra >= 0.0))jd=jd+ifrac=xtra-iEND SUBROUTINE flmoon

f90select case(nph)...case(0,2)...end select Fortran 90 includes acase construction that executes at most one of several blocks of code,depending on the value of an integer, logical, or character expression.

Ideally, thecase construction will execute more efficiently than a long sequence ofcascadedif...else if...else if... constructions.C programmers should note thatthe Fortran 90 construction, perhaps mercifully, does not haveC’s “drop-through”feature.

merge(xtra,xtra-1.0_sp, xtra >= 0.0) The merge construction in Fortran90, while intended primarily for use with vector arguments, is also a convenient wayof generating conditional scalar expressions, that is, expressions with one value, oranother, depending on the result of a logical test.

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When the arguments of amerge are vectors, parallelization by thecompiler is straightforward as an array parallel operation (see p. 964).Less obvious is how the scalar case, as above, is handled. For small-scale

parallel (SSP) machines, the natural gain is via speculative evaluation of both of thefirst two arguments simultaneously with evaluation of the test.

A good compiler should not penalize a scalar machine for use of either thescalar or vectormerge construction. The Fortran 90 standard states that “it is notnecessary for a processor to evaluate all of the operands of an expression, or toevaluate entirely each operand, if the value of the expression can be determinedotherwise.” Therefore, for each test on a scalar machine, only one or the other ofthe first two argument components need be evaluated.

� � �

FUNCTION julday(mm,id,iyyy)USE nrtype; USE nrutil, ONLY : nrerrorIMPLICIT NONEINTEGER(I4B), INTENT(IN) :: mm,id,iyyyINTEGER(I4B) :: julday

In this routine julday returns the Julian Day Number that begins at noon of the calendardate specified by month mm, day id, and year iyyy, all integer variables. Positive yearsignifies A.D.; negative, B.C. Remember that the year after 1 B.C. was 1 A.D.

INTEGER(I4B), PARAMETER :: IGREG=15+31*(10+12*1582) Gregorian Calendar adoptedOct. 15, 1582.INTEGER(I4B) :: ja,jm,jy

jy=iyyyif (jy == 0) call nrerror(’julday: there is no year zero’)if (jy < 0) jy=jy+1if (mm > 2) then Here is an example of a block IF-structure.

jm=mm+1else

jy=jy-1jm=mm+13

end ifjulday=floor(365.25_sp*jy)+floor(30.6001_sp*jm)+id+1720995if (id+31*(mm+12*iyyy) >= IGREG) then Test whether to change to Gregorian Cal-

endar.ja=floor(0.01_sp*jy)julday=julday+2-ja+floor(0.25_sp*ja)

end ifEND FUNCTION julday

� � �

PROGRAM badlukUSE nrtypeUSE nr, ONLY : flmoon,juldayIMPLICIT NONEINTEGER(I4B) :: ic,icon,idwk,ifrac,im,iyyy,jd,jday,nINTEGER(I4B) :: iybeg=1900,iyend=2000 The range of dates to be searched.REAL(SP) :: fracREAL(SP), PARAMETER :: TIMZON=-5.0_sp/24.0_spTime zone −5 is Eastern Standard Time.

write (*,’(1x,a,i5,a,i5)’) ’Full moons on Friday the 13th from’,&iybeg,’ to’,iyend

do iyyy=iybeg,iyend Loop over each year,do im=1,12 and each month.

jday=julday(im,13,iyyy) Is the 13th a Friday?idwk=mod(jday+1,7)

1012 Chapter B1. Preliminaries

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if (idwk == 5) thenn=12.37_sp*(iyyy-1900+(im-0.5_sp)/12.0_sp)This value n is a first approximation to how many full moons have occurredsince 1900. We will feed it into the phase routine and adjust it up or down untilwe determine that our desired 13th was or was not a full moon. The variableicon signals the direction of adjustment.

icon=0do

call flmoon(n,2,jd,frac) Get date of full moon n.ifrac=nint(24.0_sp*(frac+TIMZON)) Convert to hours in correct time

zone.if (ifrac < 0) thenjd=jd-1 Convert from Julian Days beginning at noon

to civil days beginning at midnight.ifrac=ifrac+24end ifif (ifrac > 12) then

jd=jd+1ifrac=ifrac-12

elseifrac=ifrac+12

end ifif (jd == jday) then Did we hit our target day?

write (*,’(/1x,i2,a,i2,a,i4)’) im,’/’,13,’/’,iyyywrite (*,’(1x,a,i2,a)’) ’Full moon ’,ifrac,&

’ hrs after midnight (EST).’Don’t worry if you are unfamiliar with FORTRAN’s esoteric input/outputstatements; very few programs in this book do any input/output.

exit Part of the break-structure, case of a match.else Didn’t hit it.

ic=isign(1,jday-jd)if (ic == -icon) exit Another break, case of no match.icon=icn=n+ic

end ifend do

end ifend do

end doEND PROGRAM badluk

f90...IGREG=15+31*(10+12*1582) (in julday), ...TIMZON=-5.0_sp/24.0_sp(in badluk) These are two examples of initialization expressions for“named constants” (that is,PARAMETERs). Because the initialization

expressions will generally be evaluated at compile time, Fortran 90 puts somerestrictions on what kinds of intrinsic functions they can contain. Although theevaluation of a real expression like-5.0_sp/24.0_sp ought to give identical resultsat compile time and at execution time, all the way down to the least significantbit, in our opinion the conservative programmer shouldn’t count on strict identityat the level of floating-point roundoff error. (In the special case ofcross-compilers,such roundoff-level discrepancies between compile time and run time are almostinevitable.)

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SUBROUTINE caldat(julian,mm,id,iyyy)USE nrtypeIMPLICIT NONEINTEGER(I4B), INTENT(IN) :: julianINTEGER(I4B), INTENT(OUT) :: mm,id,iyyy

Inverse of the function julday given above. Here julian is input as a Julian Day Number,and the routine outputs mm,id, and iyyy as the month, day, and year on which the specifiedJulian Day started at noon.

INTEGER(I4B) :: ja,jalpha,jb,jc,jd,jeINTEGER(I4B), PARAMETER :: IGREG=2299161if (julian >= IGREG) then Cross-over to Gregorian Calendar produces this

correction.jalpha=int(((julian-1867216)-0.25_dp)/36524.25_dp)ja=julian+1+jalpha-int(0.25_dp*jalpha)

else if (julian < 0) then Make day number positive by adding integer num-ber of Julian centuries, then subtract themoff at the end.

ja=julian+36525*(1-julian/36525)else

ja=julianend ifjb=ja+1524jc=int(6680.0_dp+((jb-2439870)-122.1_dp)/365.25_dp)jd=365*jc+int(0.25_dp*jc)je=int((jb-jd)/30.6001_dp)id=jb-jd-int(30.6001_dp*je)mm=je-1if (mm > 12) mm=mm-12iyyy=jc-4715if (mm > 2) iyyy=iyyy-1if (iyyy

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erica).C1. Listing of Utility Modules

(nrtype and nrutil)

C1.1 Numerical Recipes Types (nrtype)

The file supplied as nrtype.f90 contains a single module named nrtype,which in turn contains definitions for a number of named constants (that is,PARAMETERs), and a couple of elementary derived data types used by the sparsematrix routines in this book. Of the named constants, by far the most important arethose that define the KIND types of virtually all the variables used in this book: I4B,I2B, and I1B for integer variables, SP and DP for real variables (and SPC and DPC forthe corresponding complex cases), and LGT for the default logical type.

MODULE nrtypeSymbolic names for kind types of 4-, 2-, and 1-byte integers:

INTEGER, PARAMETER :: I4B = SELECTED_INT_KIND(9)INTEGER, PARAMETER :: I2B = SELECTED_INT_KIND(4)INTEGER, PARAMETER :: I1B = SELECTED_INT_KIND(2)

Symbolic names for kind types of single- and double-precision reals:INTEGER, PARAMETER :: SP = KIND(1.0)INTEGER, PARAMETER :: DP = KIND(1.0D0)

Symbolic names for kind types of single- and double-precision complex:INTEGER, PARAMETER :: SPC = KIND((1.0,1.0))INTEGER, PARAMETER :: DPC = KIND((1.0D0,1.0D0))

Symbolic name for kind type of default logical:INTEGER, PARAMETER :: LGT = KIND(.true.)

Frequently used mathematical constants (with precision to spare):REAL(SP), PARAMETER :: PI=3.141592653589793238462643383279502884197_spREAL(SP), PARAMETER :: PIO2=1.57079632679489661923132169163975144209858_spREAL(SP), PARAMETER :: TWOPI=6.283185307179586476925286766559005768394_spREAL(SP), PARAMETER :: SQRT2=1.41421356237309504880168872420969807856967_spREAL(SP), PARAMETER :: EULER=0.5772156649015328606065120900824024310422_spREAL(DP), PARAMETER :: PI_D=3.141592653589793238462643383279502884197_dpREAL(DP), PARAMETER :: PIO2_D=1.57079632679489661923132169163975144209858_dpREAL(DP), PARAMETER :: TWOPI_D=6.283185307179586476925286766559005768394_dp

Derived data types for sparse matrices, single and double precision (see use in Chapter B2):TYPE sprs2_sp

INTEGER(I4B) :: n,lenREAL(SP), DIMENSION(:), POINTER :: valINTEGER(I4B), DIMENSION(:), POINTER :: irowINTEGER(I4B), DIMENSION(:), POINTER :: jcol

END TYPE sprs2_spTYPE sprs2_dp

INTEGER(I4B) :: n,lenREAL(DP), DIMENSION(:), POINTER :: val

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1362 Appendix C1. Listing of Utility Modules (nrtype and nrutil)

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INTEGER(I4B), DIMENSION(:), POINTER :: irowINTEGER(I4B), DIMENSION(:), POINTER :: jcol

END TYPE sprs2_dpEND MODULE nrtype

About Converting to Higher Precision

You might hope that changing all the Numerical Recipes routines from singleprecision to double precision would be as simple as redefining the values of SP andDP in nrtype. Well . . . not quite.

Converting algorithms to a higher precision is not a purely mechanical taskbecause of the distinction between “roundoff error” and “truncation error.” (Pleasesee Volume 1, §1.2, if you are not familiar with these concepts.) While increasing theprecision implied by the kind values SP and DPwill indeed reduce a routine’s roundofferror, it will not reduce any truncation error that may be intrinsic to the algorithm.Sometimes, a routine contains “accuracy parameters” that can be adjusted to reducethe truncation error to the new, desired level. In other cases, however, the truncationerror cannot be so easily reduced; then, a whole new algorithm is needed. Clearlysuch new algorithms are beyond the scope of a simple mechanical “conversion.”

If, despite these cautionary words, you want to proceed with converting someroutines to a higher precision, here are some hints:

If your machine has a kind type that is distinct from, and has equal or greaterprecision than, the kind type that we use for DP, then, in nrtype, you can simplyredefine DP to this new highest precision and redefine SP to what was previouslyDP. For example, DEC machines usually have a “quadruple precision” real typeavailable, which can be used in this way. You should not need to make any furtheredits of nrtype or nrutil.

If, on the other hand, the kind type that we already use for DP is the highestprecision available, then you must leave DP defined as it is, and redefine SP in nrtypeto be this same kind type. Now, however, you will also have to edit nrutil, becausesome overloaded routines that were previously distinguishable (by the different kindtypes) will now be seen by the compiler as indistinguishable — and it will objectstrenuously. Simply delete all the “ dp” function names from the list of overloadedprocedures (i.e., from the MODULE PROCEDURE statements). Note that it is notnecessary to delete the routines from the MODULE itself. Similarly, in the interfacefile nr.f90 you must delete the “ dp” interfaces, except for the sprs... routines.(Since they have TYPE(sprs2 dp) or TYPE(sprs2 sp), they are treated as distincteven though they have functionally equivalent kind types.)

Finally, the following table gives some suggestions for changing the accuracyparameters, or constants, in some of the routines. Please note that this table is notnecessarily complete, and that higher-precision performance is not guaranteed for allthe routines, even if you make all the changes indicated. The above edits, and thesesuggestions, do, however, work in the majority of cases.

C1.1 Numerical Recipes Types (nrtype) 1363

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In routine... change... to...

beschb NUSE1=5,NUSE2=5 NUSE1=7,NUSE2=8bessi IACC=40 IACC=200bessik EPS=1.0e-10 dp EPS=epsilon(x)bessj IACC=40 IACC=160bessjy EPS=1.0e-10 dp EPS=epsilon(x)broydn TOLF=1.0e-4 sp TOLF=1.0e-8 sp

TOLMIN=1.0e-6 sp TOLMIN=1.0e-12 sp

fdjac EPS=1.0e-4 sp EPS=1.0e-8 spfrprmn EPS=1.0e-10 sp EPS=1.0e-18 spgauher EPS=3.0e-13 dp EPS=1.0e-14 dpgaujac EPS=3.0e-14 dp EPS=1.0e-14 dpgaulag EPS=3.0e-13 dp EPS=1.0e-14 dpgauleg EPS=3.0e-14 dp EPS=1.0e-14 dphypgeo EPS=1.0e-6 sp EPS=1.0e-14 splinmin TOL=1.0e-4 sp TOL=1.0e-8 spnewt TOLF=1.0e-4 sp TOLF=1.0e-8 sp

TOLMIN=1.0e-6 sp TOLMIN=1.0e-12 sp

probks EPS1=0.001 sp EPS1=1.0e-6 spEPS2=1.0e-8 sp EPS2=1.0e-16 sp

qromb EPS=1.0e-6 sp EPS=1.0e-10 spqromo EPS=1.0e-6 sp EPS=1.0e-10 spqroot TINY=1.0e-6 sp TINY=1.0e-14 spqsimp EPS=1.0e-6 sp EPS=1.0e-10 spqtrap EPS=1.0e-6 sp EPS=1.0e-10 sprc ERRTOL=0.04 sp ERRTOL=0.0012 sprd ERRTOL=0.05 sp ERRTOL=0.0015 sprf ERRTOL=0.08 sp ERRTOL=0.0025 sprj ERRTOL=0.05 sp ERRTOL=0.0015 spsfroid conv=5.0e-6 sp conv=1.0e-14 spshoot EPS=1.0e-6 sp EPS=1.0e-14 spshootf EPS=1.0e-6 sp EPS=1.0e-14 spsimplx EPS=1.0e-6 sp EPS=1.0e-14 spsncndn CA=0.0003 sp CA=1.0e-8 spsor EPS=1.0e-5 dp EPS=1.0e-13 dpsphfpt DXX=1.0e-4 sp DXX=1.0e-8 spsphoot dx=1.0e-4 sp dx=1.0e-8 spsvdfit TOL=1.0e-5 sp TOL=1.0e-13 spzroots EPS=1.0e-6 sp EPS=1.0e-14 sp


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C1.2 Numerical Recipes Utilities (nrutil)

The file supplied as nrutil.f90 contains a single module named nrutil,which contains specific implementations for all the Numerical Recipes utilityfunctions described in detail in Chapter 23.

The specific implementations given are something of a compromise betweendemonstrating parallel techniques (when they can be achieved in Fortran 90) andrunning efficiently on conventional, serial machines. The parameters at the beginningof the module (names beginning with NPAR ) are typically related to array lengthsbelow which the implementations revert to serial operations. On a purely serialmachine, these can be set to large values to suppress many parallel constructions.

The length and repetitiveness of the nrutil.f90 file stems in large part fromits extensive use of overloading. Indeed, the file would be even longer if weoverloaded versions for all the applicable data types that each utility could, inprinciple, instantiate. The descriptions in Chapter 23 detail both the full set ofintended data types and shapes for each routine, and also the types and shapesactually here implemented (which can also be gleaned by examining the file). Theintended result of all this overloading is, in essence, to give the utility routines thedesirable properties of many of the Fortran 90 intrinsic functions, namely, to beboth generic (apply to many data types) and elemental (apply element-by-elementto arbitrary shapes). Fortran 95’s provision of user-defined elemental functions willreduce the multiplicity of overloading in some of our routines; unfortunately thenecessity to overload for multiple data types will still be present.

Finally, it is worth reemphasizing the following point, already made in Chapter23: The purpose of the nrutil utilities is to remove from the Numerical Recipesprograms just those programming tasks and “idioms” whose efficient implementationis most hardware and compiler dependent, so as to allow for specific, efficientimplementations on different machines. One should therefore not expect the utmostin efficiency from the general purpose, one-size-fits-all, implementation listed here.

Correspondingly, we would encourage the incorporation of efficient nrutilimplementations, and/or comparable capabilities under different names, with asbroad as possible a set of overloaded data types, in libraries associated with specificcompilers or machines. In support of this goal, we have specifically put this AppendixC1, and the files nrtype.f90 and nrutil.f90, into the public domain.

MODULE nrutilTABLE OF CONTENTS OF THE NRUTIL MODULE:

routines that move data:array copy, swap, reallocate

routines returning a location as an integer valueifirstloc, imaxloc, iminloc

routines for argument checking and error handling:assert, assert eq, nrerror

routines relating to polynomials and recurrencesarth, geop, cumsum, cumprod, poly, polyterm,zroots unity

routines for ”outer” operations on vectorsouterand, outersum, outerdiff, outerprod, outerdiv

routines for scatter-with-combinescatter add, scatter max

routines for skew operations on matricesdiagadd, diagmult, get diag, put diag,

C1.2 Numerical Recipes Utilities (nrutil) 1365

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unit matrix, lower triangle, upper trianglemiscellaneous routines

vabsUSE nrtype

Parameters for crossover from serial to parallel algorithms (these are used only within thisnrutil module):

IMPLICIT NONEINTEGER(I4B), PARAMETER :: NPAR_ARTH=16,NPAR2_ARTH=8 Each NPAR2 must be ≤ the

corresponding NPAR.INTEGER(I4B), PARAMETER :: NPAR_GEOP=4,NPAR2_GEOP=2INTEGER(I4B), PARAMETER :: NPAR_CUMSUM=16INTEGER(I4B), PARAMETER :: NPAR_CUMPROD=8INTEGER(I4B), PARAMETER :: NPAR_POLY=8INTEGER(I4B), PARAMETER :: NPAR_POLYTERM=8

Next, generic interfaces for routines with overloaded versions. Naming conventions for ap-pended codes in the names of overloaded routines are as follows: r=real, d=double pre-cision, i=integer, c=complex, z=double-precision complex, h=character, l=logical. Any ofr,d,i,c,z,h,l may be followed by v=vector or m=matrix (v,m suffixes are used only whenneeded to resolve ambiguities).Routines that move data:

INTERFACE array_copyMODULE PROCEDURE array_copy_r, array_copy_d, array_copy_i

END INTERFACEINTERFACE swap

MODULE PROCEDURE swap_i,swap_r,swap_rv,swap_c, &swap_cv,swap_cm,swap_z,swap_zv,swap_zm, &masked_swap_rs,masked_swap_rv,masked_swap_rm

END INTERFACEINTERFACE reallocate

MODULE PROCEDURE reallocate_rv,reallocate_rm,&reallocate_iv,reallocate_im,reallocate_hv

END INTERFACERoutines returning a location as an integer value (ifirstloc, iminloc are not currently over-loaded and so do not have a generic interface here):

INTERFACE imaxlocMODULE PROCEDURE imaxloc_r,imaxloc_i

END INTERFACERoutines for argument checking and error handling (nrerror is not currently overloaded):

INTERFACE assertMODULE PROCEDURE assert1,assert2,assert3,assert4,assert_v

END INTERFACEINTERFACE assert_eq

MODULE PROCEDURE assert_eq2,assert_eq3,assert_eq4,assert_eqnEND INTERFACE

Routines relating to polynomials and recurrences (cumprod, zroots unity are not currentlyoverloaded):

INTERFACE arthMODULE PROCEDURE arth_r, arth_d, arth_i

END INTERFACEINTERFACE geop

MODULE PROCEDURE geop_r, geop_d, geop_i, geop_c, geop_dvEND INTERFACEINTERFACE cumsum

MODULE PROCEDURE cumsum_r,cumsum_iEND INTERFACEINTERFACE poly

MODULE PROCEDURE poly_rr,poly_rrv,poly_dd,poly_ddv,&poly_rc,poly_cc,poly_msk_rrv,poly_msk_ddv

END INTERFACEINTERFACE poly_term

MODULE PROCEDURE poly_term_rr,poly_term_ccEND INTERFACE

Routines for “outer” operations on vectors (outerand, outersum, outerdiv are not currentlyoverloaded):

INTERFACE outerprod


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MODULE PROCEDURE outerprod_r,outerprod_dEND INTERFACEINTERFACE outerdiff

MODULE PROCEDURE outerdiff_r,outerdiff_d,outerdiff_iEND INTERFACE

Routines for scatter-with-combine, scatter add, scatter max:INTERFACE scatter_add

MODULE PROCEDURE scatter_add_r,scatter_add_dEND INTERFACEINTERFACE scatter_max

MODULE PROCEDURE scatter_max_r,scatter_max_dEND INTERFACE

Routines for skew operations on matrices (unit matrix, lower triangle, upper triangle notcurrently overloaded):

INTERFACE diagaddMODULE PROCEDURE diagadd_rv,diagadd_r

END INTERFACEINTERFACE diagmult

MODULE PROCEDURE diagmult_rv,diagmult_rEND INTERFACEINTERFACE get_diag

MODULE PROCEDURE get_diag_rv, get_diag_dvEND INTERFACEINTERFACE put_diag

MODULE PROCEDURE put_diag_rv, put_diag_rEND INTERFACE

Other routines (vabs is not currently overloaded):CONTAINS

Routines that move data:SUBROUTINE array_copy_r(src,dest,n_copied,n_not_copied)

Copy array where size of source not known in advance.REAL(SP), DIMENSION(:), INTENT(IN) :: srcREAL(SP), DIMENSION(:), INTENT(OUT) :: destINTEGER(I4B), INTENT(OUT) :: n_copied, n_not_copiedn_copied=min(size(src),size(dest))n_not_copied=size(src)-n_copieddest(1:n_copied)=src(1:n_copied)END SUBROUTINE array_copy_r

SUBROUTINE array_copy_d(src,dest,n_copied,n_not_copied)REAL(DP), DIMENSION(:), INTENT(IN) :: srcREAL(DP), DIMENSION(:), INTENT(OUT) :: destINTEGER(I4B), INTENT(OUT) :: n_copied, n_not_copiedn_copied=min(size(src),size(dest))n_not_copied=size(src)-n_copieddest(1:n_copied)=src(1:n_copied)END SUBROUTINE array_copy_d

SUBROUTINE array_copy_i(src,dest,n_copied,n_not_copied)INTEGER(I4B), DIMENSION(:), INTENT(IN) :: srcINTEGER(I4B), DIMENSION(:), INTENT(OUT) :: destINTEGER(I4B), INTENT(OUT) :: n_copied, n_not_copiedn_copied=min(size(src),size(dest))n_not_copied=size(src)-n_copieddest(1:n_copied)=src(1:n_copied)END SUBROUTINE array_copy_i

SUBROUTINE swap_i(a,b)Swap the contents of a and b.

INTEGER(I4B), INTENT(INOUT) :: a,bINTEGER(I4B) :: dumdum=aa=bb=dumEND SUBROUTINE swap_i


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SUBROUTINE swap_r(a,b)REAL(SP), INTENT(INOUT) :: a,bREAL(SP) :: dumdum=aa=bb=dumEND SUBROUTINE swap_r

SUBROUTINE swap_rv(a,b)REAL(SP), DIMENSION(:), INTENT(INOUT) :: a,bREAL(SP), DIMENSION(SIZE(a)) :: dumdum=aa=bb=dumEND SUBROUTINE swap_rv

SUBROUTINE swap_c(a,b)COMPLEX(SPC), INTENT(INOUT) :: a,bCOMPLEX(SPC) :: dumdum=aa=bb=dumEND SUBROUTINE swap_c

SUBROUTINE swap_cv(a,b)COMPLEX(SPC), DIMENSION(:), INTENT(INOUT) :: a,bCOMPLEX(SPC), DIMENSION(SIZE(a)) :: dumdum=aa=bb=dumEND SUBROUTINE swap_cv

SUBROUTINE swap_cm(a,b)COMPLEX(SPC), DIMENSION(:,:), INTENT(INOUT) :: a,bCOMPLEX(SPC), DIMENSION(size(a,1),size(a,2)) :: dumdum=aa=bb=dumEND SUBROUTINE swap_cm

SUBROUTINE swap_z(a,b)COMPLEX(DPC), INTENT(INOUT) :: a,bCOMPLEX(DPC) :: dumdum=aa=bb=dumEND SUBROUTINE swap_z

SUBROUTINE swap_zv(a,b)COMPLEX(DPC), DIMENSION(:), INTENT(INOUT) :: a,bCOMPLEX(DPC), DIMENSION(SIZE(a)) :: dumdum=aa=bb=dumEND SUBROUTINE swap_zv

SUBROUTINE swap_zm(a,b)COMPLEX(DPC), DIMENSION(:,:), INTENT(INOUT) :: a,bCOMPLEX(DPC), DIMENSION(size(a,1),size(a,2)) :: dumdum=aa=bb=dumEND SUBROUTINE swap_zm

SUBROUTINE masked_swap_rs(a,b,mask)REAL(SP), INTENT(INOUT) :: a,bLOGICAL(LGT), INTENT(IN) :: maskREAL(SP) :: swp


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if (mask) thenswp=aa=bb=swp

end ifEND SUBROUTINE masked_swap_rs

SUBROUTINE masked_swap_rv(a,b,mask)REAL(SP), DIMENSION(:), INTENT(INOUT) :: a,bLOGICAL(LGT), DIMENSION(:), INTENT(IN) :: maskREAL(SP), DIMENSION(size(a)) :: swpwhere (mask)

swp=aa=bb=swp

end whereEND SUBROUTINE masked_swap_rv

SUBROUTINE masked_swap_rm(a,b,mask)REAL(SP), DIMENSION(:,:), INTENT(INOUT) :: a,bLOGICAL(LGT), DIMENSION(:,:), INTENT(IN) :: maskREAL(SP), DIMENSION(size(a,1),size(a,2)) :: swpwhere (mask)

swp=aa=bb=swp

end whereEND SUBROUTINE masked_swap_rm

FUNCTION reallocate_rv(p,n)Reallocate a pointer to a new size, preserving its previous contents.

REAL(SP), DIMENSION(:), POINTER :: p, reallocate_rvINTEGER(I4B), INTENT(IN) :: nINTEGER(I4B) :: nold,ierrallocate(reallocate_rv(n),stat=ierr)if (ierr /= 0) call &

nrerror(’reallocate_rv: problem in attempt to allocate memory’)if (.not. associated(p)) RETURNnold=size(p)reallocate_rv(1:min(nold,n))=p(1:min(nold,n))deallocate(p)END FUNCTION reallocate_rv

FUNCTION reallocate_iv(p,n)INTEGER(I4B), DIMENSION(:), POINTER :: p, reallocate_ivINTEGER(I4B), INTENT(IN) :: nINTEGER(I4B) :: nold,ierrallocate(reallocate_iv(n),stat=ierr)if (ierr /= 0) call &

nrerror(’reallocate_iv: problem in attempt to allocate memory’)if (.not. associated(p)) RETURNnold=size(p)reallocate_iv(1:min(nold,n))=p(1:min(nold,n))deallocate(p)END FUNCTION reallocate_iv

FUNCTION reallocate_hv(p,n)CHARACTER(1), DIMENSION(:), POINTER :: p, reallocate_hvINTEGER(I4B), INTENT(IN) :: nINTEGER(I4B) :: nold,ierrallocate(reallocate_hv(n),stat=ierr)if (ierr /= 0) call &

nrerror(’reallocate_hv: problem in attempt to allocate memory’)if (.not. associated(p)) RETURNnold=size(p)reallocate_hv(1:min(nold,n))=p(1:min(nold,n))


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deallocate(p)END FUNCTION reallocate_hv

FUNCTION reallocate_rm(p,n,m)REAL(SP), DIMENSION(:,:), POINTER :: p, reallocate_rmINTEGER(I4B), INTENT(IN) :: n,mINTEGER(I4B) :: nold,mold,ierrallocate(reallocate_rm(n,m),stat=ierr)if (ierr /= 0) call &

nrerror(’reallocate_rm: problem in attempt to allocate memory’)if (.not. associated(p)) RETURNnold=size(p,1)mold=size(p,2)reallocate_rm(1:min(nold,n),1:min(mold,m))=&

p(1:min(nold,n),1:min(mold,m))deallocate(p)END FUNCTION reallocate_rm

FUNCTION reallocate_im(p,n,m)INTEGER(I4B), DIMENSION(:,:), POINTER :: p, reallocate_imINTEGER(I4B), INTENT(IN) :: n,mINTEGER(I4B) :: nold,mold,ierrallocate(reallocate_im(n,m),stat=ierr)if (ierr /= 0) call &

nrerror(’reallocate_im: problem in attempt to allocate memory’)if (.not. associated(p)) RETURNnold=size(p,1)mold=size(p,2)reallocate_im(1:min(nold,n),1:min(mold,m))=&

p(1:min(nold,n),1:min(mold,m))deallocate(p)END FUNCTION reallocate_im

Routines returning a location as an integer value:FUNCTION ifirstloc(mask)

Index of first occurrence of .true. in a logical vector.LOGICAL(LGT), DIMENSION(:), INTENT(IN) :: maskINTEGER(I4B) :: ifirstlocINTEGER(I4B), DIMENSION(1) :: locloc=maxloc(merge(1,0,mask))ifirstloc=loc(1)if (.not. mask(ifirstloc)) ifirstloc=size(mask)+1END FUNCTION ifirstloc

FUNCTION imaxloc_r(arr)Index of maxloc on an array.

REAL(SP), DIMENSION(:), INTENT(IN) :: arrINTEGER(I4B) :: imaxloc_rINTEGER(I4B), DIMENSION(1) :: imaximax=maxloc(arr(:))imaxloc_r=imax(1)END FUNCTION imaxloc_r

FUNCTION imaxloc_i(iarr)INTEGER(I4B), DIMENSION(:), INTENT(IN) :: iarrINTEGER(I4B), DIMENSION(1) :: imaxINTEGER(I4B) :: imaxloc_iimax=maxloc(iarr(:))imaxloc_i=imax(1)END FUNCTION imaxloc_i

FUNCTION iminloc(arr)Index of minloc on an array.

REAL(SP), DIMENSION(:), INTENT(IN) :: arrINTEGER(I4B), DIMENSION(1) :: iminINTEGER(I4B) :: iminlocimin=minloc(arr(:))


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iminloc=imin(1)END FUNCTION iminloc

Routines for argument checking and error handling:SUBROUTINE assert1(n1,string)

Report and die if any logical is false (used for arg range checking).CHARACTER(LEN=*), INTENT(IN) :: stringLOGICAL, INTENT(IN) :: n1if (.not. n1) then

write (*,*) ’nrerror: an assertion failed with this tag:’, &string

STOP ’program terminated by assert1’end ifEND SUBROUTINE assert1

SUBROUTINE assert2(n1,n2,string)CHARACTER(LEN=*), INTENT(IN) :: stringLOGICAL, INTENT(IN) :: n1,n2if (.not. (n1 .and. n2)) then



SUBROUTINE assert3(n1,n2,n3,string)CHARACTER(LEN=*), INTENT(IN) :: stringLOGICAL, INTENT(IN) :: n1,n2,n3if (.not. (n1 .and. n2 .and. n3)) then



SUBROUTINE assert4(n1,n2,n3,n4,string)CHARACTER(LEN=*), INTENT(IN) :: stringLOGICAL, INTENT(IN) :: n1,n2,n3,n4if (.not. (n1 .and. n2 .and. n3 .and. n4)) then



SUBROUTINE assert_v(n,string)CHARACTER(LEN=*), INTENT(IN) :: stringLOGICAL, DIMENSION(:), INTENT(IN) :: nif (.not. all(n)) then


STOP ’program terminated by assert_v’end ifEND SUBROUTINE assert_v

FUNCTION assert_eq2(n1,n2,string)Report and die if integers not all equal (used for size checking).

CHARACTER(LEN=*), INTENT(IN) :: stringINTEGER, INTENT(IN) :: n1,n2INTEGER :: assert_eq2if (n1 == n2) then

assert_eq2=n1else

write (*,*) ’nrerror: an assert_eq failed with this tag:’, &string

STOP ’program terminated by assert_eq2’end if


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erica).

END FUNCTION assert_eq2

FUNCTION assert_eq3(n1,n2,n3,string)CHARACTER(LEN=*), INTENT(IN) :: stringINTEGER, INTENT(IN) :: n1,n2,n3INTEGER :: assert_eq3if (n1 == n2 .and. n2 == n3) then

assert_eq3=n1else


STOP ’program terminated by assert_eq3’end ifEND FUNCTION assert_eq3

FUNCTION assert_eq4(n1,n2,n3,n4,string)CHARACTER(LEN=*), INTENT(IN) :: stringINTEGER, INTENT(IN) :: n1,n2,n3,n4INTEGER :: assert_eq4if (n1 == n2 .and. n2 == n3 .and. n3 == n4) then

assert_eq4=n1else


STOP ’program terminated by assert_eq4’end ifEND FUNCTION assert_eq4

FUNCTION assert_eqn(nn,string)CHARACTER(LEN=*), INTENT(IN) :: stringINTEGER, DIMENSION(:), INTENT(IN) :: nnINTEGER :: assert_eqnif (all(nn(2:) == nn(1))) then

assert_eqn=nn(1)else


STOP ’program terminated by assert_eqn’end ifEND FUNCTION assert_eqn

SUBROUTINE nrerror(string)Report a message, then die.

CHARACTER(LEN=*), INTENT(IN) :: stringwrite (*,*) ’nrerror: ’,stringSTOP ’program terminated by nrerror’END SUBROUTINE nrerror

Routines relating to polynomials and recurrences:FUNCTION arth_r(first,increment,n)

Array function returning an arithmetic progression.REAL(SP), INTENT(IN) :: first,incrementINTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(n) :: arth_rINTEGER(I4B) :: k,k2REAL(SP) :: tempif (n > 0) arth_r(1)=firstif (n


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doif (k >= n) exitk2=k+karth_r(k+1:min(k2,n))=temp+arth_r(1:min(k,n-k))temp=temp+tempk=k2

end doend ifEND FUNCTION arth_r

FUNCTION arth_d(first,increment,n)REAL(DP), INTENT(IN) :: first,incrementINTEGER(I4B), INTENT(IN) :: nREAL(DP), DIMENSION(n) :: arth_dINTEGER(I4B) :: k,k2REAL(DP) :: tempif (n > 0) arth_d(1)=firstif (n = n) exitk2=k+karth_d(k+1:min(k2,n))=temp+arth_d(1:min(k,n-k))temp=temp+tempk=k2

end doend ifEND FUNCTION arth_d

FUNCTION arth_i(first,increment,n)INTEGER(I4B), INTENT(IN) :: first,increment,nINTEGER(I4B), DIMENSION(n) :: arth_iINTEGER(I4B) :: k,k2,tempif (n > 0) arth_i(1)=firstif (n = n) exitk2=k+karth_i(k+1:min(k2,n))=temp+arth_i(1:min(k,n-k))temp=temp+tempk=k2

end doend ifEND FUNCTION arth_i

FUNCTION geop_r(first,factor,n)Array function returning a geometric progression.

REAL(SP), INTENT(IN) :: first,factor


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INTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(n) :: geop_rINTEGER(I4B) :: k,k2REAL(SP) :: tempif (n > 0) geop_r(1)=firstif (n = n) exitk2=k+kgeop_r(k+1:min(k2,n))=temp*geop_r(1:min(k,n-k))temp=temp*tempk=k2

end doend ifEND FUNCTION geop_r

FUNCTION geop_d(first,factor,n)REAL(DP), INTENT(IN) :: first,factorINTEGER(I4B), INTENT(IN) :: nREAL(DP), DIMENSION(n) :: geop_dINTEGER(I4B) :: k,k2REAL(DP) :: tempif (n > 0) geop_d(1)=firstif (n = n) exitk2=k+kgeop_d(k+1:min(k2,n))=temp*geop_d(1:min(k,n-k))temp=temp*tempk=k2

end doend ifEND FUNCTION geop_d

FUNCTION geop_i(first,factor,n)INTEGER(I4B), INTENT(IN) :: first,factor,nINTEGER(I4B), DIMENSION(n) :: geop_iINTEGER(I4B) :: k,k2,tempif (n > 0) geop_i(1)=firstif (n


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erica).

temp=factor**NPAR2_GEOPk=NPAR2_GEOPdo

if (k >= n) exitk2=k+kgeop_i(k+1:min(k2,n))=temp*geop_i(1:min(k,n-k))temp=temp*tempk=k2

end doend ifEND FUNCTION geop_i

FUNCTION geop_c(first,factor,n)COMPLEX(SP), INTENT(IN) :: first,factorINTEGER(I4B), INTENT(IN) :: nCOMPLEX(SP), DIMENSION(n) :: geop_cINTEGER(I4B) :: k,k2COMPLEX(SP) :: tempif (n > 0) geop_c(1)=firstif (n = n) exitk2=k+kgeop_c(k+1:min(k2,n))=temp*geop_c(1:min(k,n-k))temp=temp*tempk=k2

end doend ifEND FUNCTION geop_c

FUNCTION geop_dv(first,factor,n)REAL(DP), DIMENSION(:), INTENT(IN) :: first,factorINTEGER(I4B), INTENT(IN) :: nREAL(DP), DIMENSION(size(first),n) :: geop_dvINTEGER(I4B) :: k,k2REAL(DP), DIMENSION(size(first)) :: tempif (n > 0) geop_dv(:,1)=first(:)if (n = n) exitk2=k+kgeop_dv(:,k+1:min(k2,n))=geop_dv(:,1:min(k,n-k))*&

spread(temp,2,size(geop_dv(:,1:min(k,n-k)),2))temp=temp*tempk=k2

end doend if


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orth Am

erica).

END FUNCTION geop_dv

RECURSIVE FUNCTION cumsum_r(arr,seed) RESULT(ans)Cumulative sum on an array, with optional additive seed.

REAL(SP), DIMENSION(:), INTENT(IN) :: arrREAL(SP), OPTIONAL, INTENT(IN) :: seedREAL(SP), DIMENSION(size(arr)) :: ansINTEGER(I4B) :: n,jREAL(SP) :: sdn=size(arr)if (n == 0_i4b) RETURNsd=0.0_spif (present(seed)) sd=seedans(1)=arr(1)+sdif (n < NPAR_CUMSUM) then

do j=2,nans(j)=ans(j-1)+arr(j)

end doelse

ans(2:n:2)=cumsum_r(arr(2:n:2)+arr(1:n-1:2),sd)ans(3:n:2)=ans(2:n-1:2)+arr(3:n:2)

end ifEND FUNCTION cumsum_r

RECURSIVE FUNCTION cumsum_i(arr,seed) RESULT(ans)INTEGER(I4B), DIMENSION(:), INTENT(IN) :: arrINTEGER(I4B), OPTIONAL, INTENT(IN) :: seedINTEGER(I4B), DIMENSION(size(arr)) :: ansINTEGER(I4B) :: n,j,sdn=size(arr)if (n == 0_i4b) RETURNsd=0_i4bif (present(seed)) sd=seedans(1)=arr(1)+sdif (n < NPAR_CUMSUM) then

do j=2,nans(j)=ans(j-1)+arr(j)

end doelse

ans(2:n:2)=cumsum_i(arr(2:n:2)+arr(1:n-1:2),sd)ans(3:n:2)=ans(2:n-1:2)+arr(3:n:2)

end ifEND FUNCTION cumsum_i

RECURSIVE FUNCTION cumprod(arr,seed) RESULT(ans)Cumulative product on an array, with optional multiplicative seed.

REAL(SP), DIMENSION(:), INTENT(IN) :: arrREAL(SP), OPTIONAL, INTENT(IN) :: seedREAL(SP), DIMENSION(size(arr)) :: ansINTEGER(I4B) :: n,jREAL(SP) :: sdn=size(arr)if (n == 0_i4b) RETURNsd=1.0_spif (present(seed)) sd=seedans(1)=arr(1)*sdif (n < NPAR_CUMPROD) then

do j=2,nans(j)=ans(j-1)*arr(j)

end doelse

ans(2:n:2)=cumprod(arr(2:n:2)*arr(1:n-1:2),sd)ans(3:n:2)=ans(2:n-1:2)*arr(3:n:2)

end ifEND FUNCTION cumprod


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FUNCTION poly_rr(x,coeffs)Polynomial evaluation.

REAL(SP), INTENT(IN) :: xREAL(SP), DIMENSION(:), INTENT(IN) :: coeffsREAL(SP) :: poly_rrREAL(SP) :: powREAL(SP), DIMENSION(:), ALLOCATABLE :: vecINTEGER(I4B) :: i,n,nnn=size(coeffs)if (n


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erica).

FUNCTION poly_rc(x,coeffs)COMPLEX(SPC), INTENT(IN) :: xREAL(SP), DIMENSION(:), INTENT(IN) :: coeffsCOMPLEX(SPC) :: poly_rcCOMPLEX(SPC) :: powCOMPLEX(SPC), DIMENSION(:), ALLOCATABLE :: vecINTEGER(I4B) :: i,n,nnn=size(coeffs)if (n


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erica).

REAL(SP), DIMENSION(:), INTENT(IN) :: coeffs,xREAL(SP), DIMENSION(size(x)) :: poly_rrvINTEGER(I4B) :: i,n,mm=size(coeffs)n=size(x)if (m


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erica).

elseu(2:n:2)=poly_term_rr(a(2:n:2)+a(1:n-1:2)*b,b*b)u(3:n:2)=a(3:n:2)+b*u(2:n-1:2)

end ifEND FUNCTION poly_term_rr

RECURSIVE FUNCTION poly_term_cc(a,b) RESULT(u)COMPLEX(SPC), DIMENSION(:), INTENT(IN) :: aCOMPLEX(SPC), INTENT(IN) :: bCOMPLEX(SPC), DIMENSION(size(a)) :: uINTEGER(I4B) :: n,jn=size(a)if (n = nn) exitzroots_unity(k+1)=cmplx(cos(k*theta),sin(k*theta),SPC)zroots_unity(k+2:min(2*k,nn))=zroots_unity(k+1)*&

zroots_unity(2:min(k,nn-k))k=2*k

end doEND FUNCTION zroots_unity

Routines for “outer” operations on vectors. The order convention is: result(i,j) = first operand(i)(op) second operand(j).

FUNCTION outerprod_r(a,b)REAL(SP), DIMENSION(:), INTENT(IN) :: a,bREAL(SP), DIMENSION(size(a),size(b)) :: outerprod_routerprod_r = spread(a,dim=2,ncopies=size(b)) * &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerprod_r

FUNCTION outerprod_d(a,b)REAL(DP), DIMENSION(:), INTENT(IN) :: a,bREAL(DP), DIMENSION(size(a),size(b)) :: outerprod_douterprod_d = spread(a,dim=2,ncopies=size(b)) * &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerprod_d

FUNCTION outerdiv(a,b)REAL(SP), DIMENSION(:), INTENT(IN) :: a,bREAL(SP), DIMENSION(size(a),size(b)) :: outerdivouterdiv = spread(a,dim=2,ncopies=size(b)) / &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerdiv

FUNCTION outersum(a,b)REAL(SP), DIMENSION(:), INTENT(IN) :: a,b


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erica).

REAL(SP), DIMENSION(size(a),size(b)) :: outersumoutersum = spread(a,dim=2,ncopies=size(b)) + &

spread(b,dim=1,ncopies=size(a))END FUNCTION outersum

FUNCTION outerdiff_r(a,b)REAL(SP), DIMENSION(:), INTENT(IN) :: a,bREAL(SP), DIMENSION(size(a),size(b)) :: outerdiff_routerdiff_r = spread(a,dim=2,ncopies=size(b)) - &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerdiff_r

FUNCTION outerdiff_d(a,b)REAL(DP), DIMENSION(:), INTENT(IN) :: a,bREAL(DP), DIMENSION(size(a),size(b)) :: outerdiff_douterdiff_d = spread(a,dim=2,ncopies=size(b)) - &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerdiff_d

FUNCTION outerdiff_i(a,b)INTEGER(I4B), DIMENSION(:), INTENT(IN) :: a,bINTEGER(I4B), DIMENSION(size(a),size(b)) :: outerdiff_iouterdiff_i = spread(a,dim=2,ncopies=size(b)) - &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerdiff_i

FUNCTION outerand(a,b)LOGICAL(LGT), DIMENSION(:), INTENT(IN) :: a,bLOGICAL(LGT), DIMENSION(size(a),size(b)) :: outerandouterand = spread(a,dim=2,ncopies=size(b)) .and. &

spread(b,dim=1,ncopies=size(a))END FUNCTION outerand

Routines for scatter-with-combine.SUBROUTINE scatter_add_r(dest,source,dest_index)REAL(SP), DIMENSION(:), INTENT(OUT) :: destREAL(SP), DIMENSION(:), INTENT(IN) :: sourceINTEGER(I4B), DIMENSION(:), INTENT(IN) :: dest_indexINTEGER(I4B) :: m,n,j,in=assert_eq2(size(source),size(dest_index),’scatter_add_r’)m=size(dest)do j=1,n

i=dest_index(j)if (i > 0 .and. i 0 .and. i


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if (i > 0 .and. i 0 .and. i


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erica).

end doEND FUNCTION get_diag_rv

FUNCTION get_diag_dv(mat)REAL(DP), DIMENSION(:,:), INTENT(IN) :: matREAL(DP), DIMENSION(size(mat,1)) :: get_diag_dvINTEGER(I4B) :: jj=assert_eq2(size(mat,1),size(mat,2),’get_diag_dv’)do j=1,size(mat,1)

get_diag_dv(j)=mat(j,j)end doEND FUNCTION get_diag_dv

SUBROUTINE put_diag_rv(diagv,mat)Set the diagonal of matrix mat to the values of a vector or scalar.

REAL(SP), DIMENSION(:), INTENT(IN) :: diagvREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: matINTEGER(I4B) :: j,nn=assert_eq2(size(diagv),min(size(mat,1),size(mat,2)),’put_diag_rv’)do j=1,n

mat(j,j)=diagv(j)end doEND SUBROUTINE put_diag_rv

SUBROUTINE put_diag_r(scal,mat)REAL(SP), INTENT(IN) :: scalREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: matINTEGER(I4B) :: j,nn = min(size(mat,1),size(mat,2))do j=1,n

mat(j,j)=scalend doEND SUBROUTINE put_diag_r

SUBROUTINE unit_matrix(mat)Set the matrix mat to be a unit matrix (if it is square).

REAL(SP), DIMENSION(:,:), INTENT(OUT) :: matINTEGER(I4B) :: i,nn=min(size(mat,1),size(mat,2))mat(:,:)=0.0_spdo i=1,n

mat(i,i)=1.0_spend doEND SUBROUTINE unit_matrix

FUNCTION upper_triangle(j,k,extra)Return an upper triangular logical mask.

INTEGER(I4B), INTENT(IN) :: j,kINTEGER(I4B), OPTIONAL, INTENT(IN) :: extraLOGICAL(LGT), DIMENSION(j,k) :: upper_triangleINTEGER(I4B) :: nn=0if (present(extra)) n=extraupper_triangle=(outerdiff(arth_i(1,1,j),arth_i(1,1,k)) < n)END FUNCTION upper_triangle

FUNCTION lower_triangle(j,k,extra)Return a lower triangular logical mask.

INTEGER(I4B), INTENT(IN) :: j,kINTEGER(I4B), OPTIONAL, INTENT(IN) :: extraLOGICAL(LGT), DIMENSION(j,k) :: lower_triangleINTEGER(I4B) :: nn=0if (present(extra)) n=extralower_triangle=(outerdiff(arth_i(1,1,j),arth_i(1,1,k)) > -n)END FUNCTION lower_triangle

Other routines:


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erica).

FUNCTION vabs(v)Return the length (ordinary L2 norm) of a vector.

REAL(SP), DIMENSION(:), INTENT(IN) :: vREAL(SP) :: vabsvabs=sqrt(dot_product(v,v))END FUNCTION vabs

END MODULE nrutil

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erica).C2. Alphabetical Listing of

Explicit Interfaces

The file supplied as nr.f90 contains explicit interfaces for all the NumericalRecipes routines (except those already in the module nrutil). The interfaces arein alphabetical order, by the generic interface name, if one exists, or by the specificroutine name if there is no generic name.

The file nr.f90 is normally invoked via a USE statement within a main programor subroutine that references a Numerical Recipes routine. See §21.1 for an example.

MODULE nrINTERFACE

SUBROUTINE airy(x,ai,bi,aip,bip)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP), INTENT(OUT) :: ai,bi,aip,bipEND SUBROUTINE airy

END INTERFACEINTERFACE

SUBROUTINE amebsa(p,y,pb,yb,ftol,func,iter,temptr)USE nrtypeINTEGER(I4B), INTENT(INOUT) :: iterREAL(SP), INTENT(INOUT) :: ybREAL(SP), INTENT(IN) :: ftol,temptrREAL(SP), DIMENSION(:), INTENT(INOUT) :: y,pbREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: pINTERFACE

FUNCTION func(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP) :: funcEND FUNCTION func

END INTERFACEEND SUBROUTINE amebsa


SUBROUTINE amoeba(p,y,ftol,func,iter)USE nrtypeINTEGER(I4B), INTENT(OUT) :: iterREAL(SP), INTENT(IN) :: ftolREAL(SP), DIMENSION(:), INTENT(INOUT) :: yREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: pINTERFACE

FUNCTION func(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP) :: funcEND FUNCTION func

END INTERFACE

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erica).

END SUBROUTINE amoebaEND INTERFACEINTERFACE

SUBROUTINE anneal(x,y,iorder)USE nrtypeINTEGER(I4B), DIMENSION(:), INTENT(INOUT) :: iorderREAL(SP), DIMENSION(:), INTENT(IN) :: x,yEND SUBROUTINE anneal


SUBROUTINE asolve(b,x,itrnsp)USE nrtypeREAL(DP), DIMENSION(:), INTENT(IN) :: bREAL(DP), DIMENSION(:), INTENT(OUT) :: xINTEGER(I4B), INTENT(IN) :: itrnspEND SUBROUTINE asolve


SUBROUTINE atimes(x,r,itrnsp)USE nrtypeREAL(DP), DIMENSION(:), INTENT(IN) :: xREAL(DP), DIMENSION(:), INTENT(OUT) :: rINTEGER(I4B), INTENT(IN) :: itrnspEND SUBROUTINE atimes


SUBROUTINE avevar(data,ave,var)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: dataREAL(SP), INTENT(OUT) :: ave,varEND SUBROUTINE avevar


SUBROUTINE balanc(a)USE nrtypeREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: aEND SUBROUTINE balanc


SUBROUTINE banbks(a,m1,m2,al,indx,b)USE nrtypeINTEGER(I4B), INTENT(IN) :: m1,m2INTEGER(I4B), DIMENSION(:), INTENT(IN) :: indxREAL(SP), DIMENSION(:,:), INTENT(IN) :: a,alREAL(SP), DIMENSION(:), INTENT(INOUT) :: bEND SUBROUTINE banbks


SUBROUTINE bandec(a,m1,m2,al,indx,d)USE nrtypeINTEGER(I4B), INTENT(IN) :: m1,m2INTEGER(I4B), DIMENSION(:), INTENT(OUT) :: indxREAL(SP), INTENT(OUT) :: dREAL(SP), DIMENSION(:,:), INTENT(INOUT) :: aREAL(SP), DIMENSION(:,:), INTENT(OUT) :: alEND SUBROUTINE bandec


SUBROUTINE banmul(a,m1,m2,x,b)USE nrtypeINTEGER(I4B), INTENT(IN) :: m1,m2REAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(:), INTENT(OUT) :: bREAL(SP), DIMENSION(:,:), INTENT(IN) :: a

1386 Appendix C2. Alphabetical Listing of Explicit Interfaces

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END SUBROUTINE banmulEND INTERFACEINTERFACE

SUBROUTINE bcucof(y,y1,y2,y12,d1,d2,c)USE nrtypeREAL(SP), INTENT(IN) :: d1,d2REAL(SP), DIMENSION(4), INTENT(IN) :: y,y1,y2,y12REAL(SP), DIMENSION(4,4), INTENT(OUT) :: cEND SUBROUTINE bcucof


SUBROUTINE bcuint(y,y1,y2,y12,x1l,x1u,x2l,x2u,x1,x2,ansy,&ansy1,ansy2)

USE nrtypeREAL(SP), DIMENSION(4), INTENT(IN) :: y,y1,y2,y12REAL(SP), INTENT(IN) :: x1l,x1u,x2l,x2u,x1,x2REAL(SP), INTENT(OUT) :: ansy,ansy1,ansy2END SUBROUTINE bcuint

END INTERFACEINTERFACE beschb

SUBROUTINE beschb_s(x,gam1,gam2,gampl,gammi)USE nrtypeREAL(DP), INTENT(IN) :: xREAL(DP), INTENT(OUT) :: gam1,gam2,gampl,gammiEND SUBROUTINE beschb_s

SUBROUTINE beschb_v(x,gam1,gam2,gampl,gammi)USE nrtypeREAL(DP), DIMENSION(:), INTENT(IN) :: xREAL(DP), DIMENSION(:), INTENT(OUT) :: gam1,gam2,gampl,gammiEND SUBROUTINE beschb_v

END INTERFACEINTERFACE bessi

FUNCTION bessi_s(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), INTENT(IN) :: xREAL(SP) :: bessi_sEND FUNCTION bessi_s

FUNCTION bessi_v(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessi_vEND FUNCTION bessi_v

END INTERFACEINTERFACE bessi0

FUNCTION bessi0_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessi0_sEND FUNCTION bessi0_s

FUNCTION bessi0_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessi0_vEND FUNCTION bessi0_v

END INTERFACEINTERFACE bessi1

FUNCTION bessi1_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessi1_sEND FUNCTION bessi1_s


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erica).

FUNCTION bessi1_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessi1_vEND FUNCTION bessi1_v


SUBROUTINE bessik(x,xnu,ri,rk,rip,rkp)USE nrtypeREAL(SP), INTENT(IN) :: x,xnuREAL(SP), INTENT(OUT) :: ri,rk,rip,rkpEND SUBROUTINE bessik

END INTERFACEINTERFACE bessj

FUNCTION bessj_s(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), INTENT(IN) :: xREAL(SP) :: bessj_sEND FUNCTION bessj_s

FUNCTION bessj_v(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessj_vEND FUNCTION bessj_v

END INTERFACEINTERFACE bessj0

FUNCTION bessj0_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessj0_sEND FUNCTION bessj0_s

FUNCTION bessj0_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessj0_vEND FUNCTION bessj0_v

END INTERFACEINTERFACE bessj1

FUNCTION bessj1_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessj1_sEND FUNCTION bessj1_s

FUNCTION bessj1_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessj1_vEND FUNCTION bessj1_v

END INTERFACEINTERFACE bessjy

SUBROUTINE bessjy_s(x,xnu,rj,ry,rjp,ryp)USE nrtypeREAL(SP), INTENT(IN) :: x,xnuREAL(SP), INTENT(OUT) :: rj,ry,rjp,rypEND SUBROUTINE bessjy_s

SUBROUTINE bessjy_v(x,xnu,rj,ry,rjp,ryp)USE nrtypeREAL(SP), INTENT(IN) :: xnuREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(:), INTENT(OUT) :: rj,rjp,ry,ryp


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cientific Com

puting (ISB

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Copyright (C

) 1986-1996 by Cam

bridge University P

ress.Program

s Copyright (C

) 1986-1996 by Num

erical Recipes S

oftware.

Perm





o order Num


or CD

RO

Ms, visit w

ebsitehttp://w

ww

.nr.com or call 1-800-872-7423 (N

orth Am



orth Am

erica).

END SUBROUTINE bessjy_vEND INTERFACEINTERFACE bessk

FUNCTION bessk_s(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), INTENT(IN) :: xREAL(SP) :: bessk_sEND FUNCTION bessk_s

FUNCTION bessk_v(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessk_vEND FUNCTION bessk_v

END INTERFACEINTERFACE bessk0

FUNCTION bessk0_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessk0_sEND FUNCTION bessk0_s

FUNCTION bessk0_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessk0_vEND FUNCTION bessk0_v

END INTERFACEINTERFACE bessk1

FUNCTION bessk1_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessk1_sEND FUNCTION bessk1_s

FUNCTION bessk1_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessk1_vEND FUNCTION bessk1_v

END INTERFACEINTERFACE bessy

FUNCTION bessy_s(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), INTENT(IN) :: xREAL(SP) :: bessy_sEND FUNCTION bessy_s

FUNCTION bessy_v(n,x)USE nrtypeINTEGER(I4B), INTENT(IN) :: nREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessy_vEND FUNCTION bessy_v

END INTERFACEINTERFACE bessy0

FUNCTION bessy0_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessy0_sEND FUNCTION bessy0_s

FUNCTION bessy0_v(x)USE nrtype


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LLEL S

cientific Com

puting (ISB

N 0-521-57439-0)

Copyright (C

) 1986-1996 by Cam

bridge University P

ress.Program

s Copyright (C

) 1986-1996 by Num

erical Recipes S

oftware.

Perm





o order Num


or CD

RO

Ms, visit w

ebsitehttp://w

ww

.nr.com or call 1-800-872-7423 (N

orth Am



orth Am

erica).

REAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessy0_vEND FUNCTION bessy0_v

END INTERFACEINTERFACE bessy1

FUNCTION bessy1_s(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: bessy1_sEND FUNCTION bessy1_s

FUNCTION bessy1_v(x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: xREAL(SP), DIMENSION(size(x)) :: bessy1_vEND FUNCTION bessy1_v

END INTERFACEINTERFACE beta

FUNCTION beta_s(z,w)USE nrtypeREAL(SP), INTENT(IN) :: z,wREAL(SP) :: beta_sEND FUNCTION beta_s

FUNCTION beta_v(z,w)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: z,wREAL(SP), DIMENSION(size(z)) :: beta_vEND FUNCTION beta_v

END INTERFACEINTERFACE betacf

FUNCTION betacf_s(a,b,x)USE nrtypeREAL(SP), INTENT(IN) :: a,b,xREAL(SP) :: betacf_sEND FUNCTION betacf_s

FUNCTION betacf_v(a,b,x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: a,b,xREAL(SP), DIMENSION(size(x)) :: betacf_vEND FUNCTION betacf_v

END INTERFACEINTERFACE betai

FUNCTION betai_s(a,b,x)USE nrtypeREAL(SP), INTENT(IN) :: a,b,xREAL(SP) :: betai_sEND FUNCTION betai_s

FUNCTION betai_v(a,b,x)USE nrtypeREAL(SP), DIMENSION(:), INTENT(IN) :: a,b,xREAL(SP), DIMENSION(size(a)) :: betai_vEND FUNCTION betai_v

END INTERFACEINTERFACE bico

FUNCTION bico_s(n,k)USE nrtypeINTEGER(I4B), INTENT(IN) :: n,kREAL(SP) :: bico_sEND FUNCTION bico_s

FUNCTION bico_v(n,k)USE nrtypeINTEGER(I4B), DIMENSION(:), INTENT(IN) :: n,kREAL(SP), DIMENSION(size(n)) :: bico_v


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LLEL S

cientific Com

puting (ISB

N 0-521-57439-0)

Copyright (C

) 1986-1996 by Cam

bridge University P

ress.Program

s Copyright (C

) 1986-1996 by Num

erical Recipes S

oftware.

Perm





o order Num


or CD

RO

Ms, visit w

ebsitehttp://w

ww

.nr.com or call 1-800-872-7423 (N

orth Am



orth Am

erica).

END FUNCTION bico_vEND INTERFACEINTERFACE

FUNCTION bnldev(pp,n)USE nrtypeREAL(SP), INTENT(IN) :: ppINTEGER(I4B), INTENT(IN) :: nREAL(SP) :: bnldevEND FUNCTION bnldev


FUNCTION brent(ax,bx,cx,func,tol,xmin)USE nrtypeREAL(SP), INTENT(IN) :: ax,bx,cx,tolREAL(SP), INTENT(OUT) :: xminREAL(SP) :: brentINTERFACE

FUNCTION func(x)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP) :: funcEND FUNCTION func

END INTERFACEEND FUNCTION brent


SUBROUTINE broydn(x,check)USE nrtypeREAL(SP), DIMENSION(:), INTENT(INOUT) :: xLOGICAL(LGT), INTENT(OUT) :: checkEND SUBROUTINE broydn


SUBROUTINE bsstep(y,dydx,x,htry,eps,yscal,hdid,hnext,derivs)USE nrtypeREAL(SP), DIMENSION(:), INTENT(INOUT) :: yREAL(SP), DIMENSION(:), INTENT(IN) :: dydx,yscalREAL(SP), INTENT(INOUT) :: xREAL(SP), INTENT(IN) :: htry,epsREAL(SP), INTENT(OUT) :: hdid,hnextINTERFACE

SUBROUTINE derivs(x,y,dydx)USE nrtypeREAL(SP), INTENT(IN) :: xREAL(SP), DIMENSION(:), INTENT(IN) :: yREAL(SP), DIMENSION(:), INTENT(OUT) :: dydxEND SUBROUTINE derivs

END INTERFACEEND SUBROUTINE bsstep


SUBROUTINE caldat(julian,mm,id,iyyy)USE nrtypeINTEGER(I4B), INTENT(IN) :: julianINTEGER(I4B), INTENT(OUT) :: mm,id,iyyyEND SUBROUTINE caldat


FUNCTION chder(a,b,c)USE nrtypeREAL(SP), INTENT(IN) :: a,bREAL(SP), DIMENSION(:), INTENT(IN) :: cREAL(SP), DIMENSION(size(c)) :: chderEND FUNCTION chder


Sam

ple page from N

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ER

ICA

L RE

CIP

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IN F

OR

TR

AN

90: TH

E A

rt of PA

RA

LLEL S

cientific Com

puting (ISB

N 0-521-57439-0)

Copyright (C

) 1986-1996 by Cam

bridge University P

ress.Program

s Copyright (C

) 1986-1996 by Num

erical Recipes S

oftware.

Perm





o order Num


or CD

RO

Ms, visit w

ebsitehttp://w

ww

.nr.com or call 1-800-872-7423 (N

orth Am

e

fortran 90 code chapters b1–b20 - pudn.comread.pudn.com/downloads174/ebook/810137/numerical...

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