MINUIT 在 local minimum 面前束手无策
设计一个0-1之间的有很多局域最小值的快速震荡函数
FVAL = cos(100*3.1415926535*xval(1))/(1+scale*(xval(1)-0.55)**2)
scale是一个放大参数，scale越大，最小值与其它极值点差异越大
代码：
program main
implicit none
integer,parameter :: NER=kind(0.0d0)
integer :: NPAR=1,IFLAG
real(NER) :: XVAL(1)=0.500000,GRAD(1)=0
real(NER) :: FVAL
integer :: errflag,ivarbl
real :: argmn(5)=0
integer :: nn
real(NER) :: step=0.001_NER
real(NER) :: limit=0.0_NER,limit1=0.0,limit2=1.0
character(len=5) :: paraname(1)
external futil
external fcn
external MNINIT, MNEXCM, MNPARM, MNSETI, MNPOUT
call fcn(NPAR, GRAD, FVAL, XVAL, IFLAG, FUTIL)
paraname(1)= "para1"
call MNINIT (5, 6, 7)
call MNSETI ('FCN')
argmn(1) = -1.0_NER
call MNEXCM (FCN, 'SET PRINTOUT', argmn, 1, errflag, FUTIL)
CALL MNPARM(1,paraname(1),xval(1),step,limit1,limit2,errflag)
argmn(1) = 2.
call MNEXCM (FCN, 'SET STRATEGY', argmn, 1, errflag, FUTIL)
argmn(1) = 100000.
argmn(2) = 0.001
call MNEXCM(fcn, 'migrad',argmn,2,errflag,futil)
argmn(1) = 1000000.
call MNEXCM(fcn, 'improve',argmn,1,errflag,futil)
end program
function FUTIL()
integer futil
futil = 1
write(*,*) "hello world"
end function
subroutine fcn(NPAR, GRAD, FVAL, XVAL, IFLAG, FUTIL)
integer,parameter :: NER=kind(0.0D0)
integer, intent(in) :: NPAR, IFLAG
real(NER), intent(in) :: XVAL(1:NPAR)
real(NER), intent(out) :: FVAL
real(NER), intent(out) :: GRAD(1:NPAR)
integer :: scale = 20000
external :: FUTIL
if(xval(1)<=1.0 .and. xval(1)>0.0) then
FVAL = cos(100*3.1415926535*xval(1))/(1+scale*(xval(1)-0.55)**2)
else
fval =1
end if
print*,"XVAL: ",xval(1)," FVAL: ",fval
end subroutine
运行结果如下，成功找到最小值点0.55
XVAL: 0.50000000000000000 FVAL: 1.9607833970753514E-002
**********
** 1 **SET PRINTOUT 0.1587-313
**********
PARAMETER DEFINITIONS:
NO. NAME VALUE STEP SIZE LIMITS
1 'para1 ' 0.50000 0.10000E-02 0.0000 1.0000
**********
** 2 **SET STRATEGY 0.5305-314
**********
**********
** 3 **MIGRAD 0.7703E-26 0.000
**********
XVAL: 0.50000000000000000 FVAL: 1.9607833970753514E-002
XVAL: 0.50001000000696405 FVAL: 1.9615428588068692E-002
XVAL: 0.49998999999303595 FVAL: 1.9600050324587299E-002
XVAL: 0.31247240058968401 FVAL: -6.3150158445647933E-004
XVAL: 0.38955528565784270 FVAL: -1.9196591888349791E-003
XVAL: 0.35752386430665556 FVAL: 9.6016976640166780E-004
XVAL: 0.99586100702573488 FVAL: 6.7192160995500531E-005
XVAL: 0.78999910445346644 FVAL: -8.6730920852707812E-004
XVAL: 0.56478195138379472 FVAL: 1.2745559692366321E-002
XVAL: 0.47680903093615801 FVAL: 4.9771636683686926E-003
XVAL: 0.43292163144366919 FVAL: -2.2079883619285450E-003
XVAL: 0.46721284268659191 FVAL: -4.6390212924105247E-003
XVAL: 0.46731262818000863 FVAL: -4.8226003476060314E-003
XVAL: 0.46711305850466328 FVAL: -4.4517603069725272E-003
XVAL: 0.46724883567244496 FVAL: -4.7056741382685008E-003
XVAL: 0.46717684987137420 FVAL: -4.5718893726272736E-003
XVAL: 0.49038385354493741 FVAL: -1.3772385763266325E-002
XVAL: 0.48191657713136388 FVAL: 8.7948583902536168E-003
XVAL: 0.60555290644860427 FVAL: -2.7555036501283078E-003
XVAL: 0.53481623094724329 FVAL: -1.0283036798199207E-002
XVAL: 0.50699213850467006 FVAL: -1.5418088047477192E-002
XVAL: 0.50702839753009599 FVAL: -1.5685815259360688E-002
XVAL: 0.50695587944246623 FVAL: -1.5149219603553915E-002
XVAL: 0.50701575615666994 FVAL: -1.5592606588012009E-002
XVAL: 0.50696852083706656 FVAL: -1.5243085317500382E-002
MINUIT WARNING IN MIGRAD
============== Negative diagonal element 1 in Error Matrix
MINUIT WARNING IN MIGRAD
============== 0.11E+01 added to diagonal of error matrix
XVAL: 0.97870485833656828 FVAL: 2.4977501126080326E-004
XVAL: 0.90281237195628938 FVAL: 2.5473291298494033E-004
XVAL: 0.72789003164292043 FVAL: -1.2434735250361514E-003
XVAL: 0.61968485287339115 FVAL: 1.0141743611477631E-002
XVAL: 0.56208823619760873 FVAL: 0.20201709331547443
XVAL: 0.52501791104384554 FVAL: -4.1758890352707454E-004
XVAL: 0.51509679970504163 FVAL: 1.1988662909358631E-003
XVAL: 0.51016569801979084 FVAL: -3.0506537093079350E-002
XVAL: 0.51018948789178897 FVAL: -3.0529145878207011E-002
XVAL: 0.51014190812476978 FVAL: -3.0482257461208851E-002
XVAL: 0.51017859934475052 FVAL: -3.0519005598981710E-002
XVAL: 0.51015279668806024 FVAL: -3.0493577202016911E-002
XVAL: 0.51065320721730667 FVAL: -3.0629305407304800E-002
XVAL: 0.51049312011095083 FVAL: -3.0668830946729807E-002
XVAL: 0.51050602225930919 FVAL: -3.0668571609290703E-002
XVAL: 0.51048021795560250 FVAL: -3.0668580067472317E-002
MIGRAD MINIMIZATION HAS CONVERGED.
FCN= -0.3066883E-01 FROM MIGRAD STATUS=CONVERGED 40 CALLS 41 TOTAL
EDM= 0.52E-08 STRATEGY= 0 NO ERROR MATRIX
EXT PARAMETER CURRENT GUESS STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 para1 0.51049 0.10000E-02 0.65498E-03 0.16385E-03
**********
** 4 **IMPROVE 0.7703E-26
**********
XVAL: 0.51049312011095083 FVAL: -3.0668830946729807E-002
XVAL: 0.51050602225930919 FVAL: -3.0668571609290703E-002
XVAL: 0.51048021795560250 FVAL: -3.0668580067472317E-002
START ATTEMPT NO. 1 TO FIND NEW MINIMUM
XVAL: 0.49670703728104204 FVAL: 8.8394581313809475E-003
XVAL: 0.50483535344822172 FVAL: 1.2369141868642239E-003
XVAL: 0.48857959137384666 FVAL: -1.1799703623788189E-002
XVAL: 0.48045516363766605 FVAL: 1.0127864865538478E-002
XVAL: 0.46422394976684306 FVAL: 1.6293725107443173E-003
XVAL: 0.44803055279286996 FVAL: -3.8985493829021595E-003
XVAL: 0.41582562006824370 FVAL: 7.1036207309076453E-004
XVAL: 0.38397649773883230 FVAL: 5.7222903598666596E-004
XVAL: 0.32188212411925721 FVAL: 7.9694622910272278E-004
XVAL: 0.26279622971041716 FVAL: 3.8670089826211949E-004
XVAL: 0.15757414147132193 FVAL: 2.3479172797228102E-004
XVAL: 7.5389180375027676E-002 FVAL: 2.7065629817364869E-005
XVAL: 3.2519474849773689E-004 FVAL: 1.6459478936350073E-004
XVAL: 5.7464619599499833E-002 FVAL: 1.4408192179233005E-004
XVAL: 0.47690379671620392 FVAL: 5.2203402377533027E-003
XVAL: 0.91916926580119762 FVAL: 3.5432265691915965E-004
XVAL: 0.53322051510158308 FVAL: -7.9983248842938004E-002
XVAL: 4.8391212093156999E-002 FVAL: -1.7383943161937128E-004
XVAL: 0.55342398818300764 FVAL: -0.38488774603948750
XVAL: 0.89570239925297757 FVAL: 9.1537159974782785E-005
XVAL: 0.59353418304050742 FVAL: -1.1422666986878421E-002
XVAL: 1.9657690306870756E-002 FVAL: 1.7671121849950990E-004
XVAL: 0.67164909955098062 FVAL: -2.9254458381214217E-003
XVAL: 0.77760069827685485 FVAL: 7.0307884988022706E-004
XVAL: 0.81241932008156303 FVAL: -5.2587147791436294E-004
XVAL: 3.3414235885206311E-002 FVAL: -8.9511073503593448E-005
XVAL: 0.98484049562592713 FVAL: 1.3239932830549670E-005
XVAL: 0.17431477179515725 FVAL: -7.5646958754691686E-005
XVAL: 0.74829961782830723 FVAL: -1.0929938864204991E-003
XVAL: 0.99210470936737183 FVAL: -2.0184585111435944E-004
XVAL: 6.2537632077579342E-002 FVAL: 1.4698927296743087E-004
XVAL: 0.68630447494081448 FVAL: -1.0694170825791163E-003
XVAL: 8.3335147765358331E-002 FVAL: 1.1465702510367926E-004
XVAL: 0.14497635213104126 FVAL: 2.2634187451940946E-006
XVAL: 0.96553083382499683 FVAL: -4.8055819585107544E-005
XVAL: 2.9344432382344410E-004 FVAL: 1.6473588537365625E-004
XVAL: 0.16991646946296318 FVAL: -3.4586932457298817E-004
XVAL: 0.54715576626680396 FVAL: -0.53938783279209379
AN IMPROVEMENT ON THE PREVIOUS MINIMUM HAS BEEN FOUND
XVAL: 0.54715576626680396 FVAL: -0.53938783279209379
START SIMPLEX MINIMIZATION. CONVERGENCE WHEN EDM .LT. 0.10E+00
XVAL: 0.52157141090689907 FVAL: 5.1305740783478090E-002
XVAL: 0.55735853415746761 FVAL: 0.32405639078885340
XVAL: 0.54306885969492669 FVAL: 0.29076907538128022
XVAL: 0.55123949644654313 FVAL: -0.89755730531967570
XVAL: 0.55531977515583542 FVAL: 6.4045234013332911E-002
XVAL: 0.55531977515583542 FVAL: 6.4045234013332911E-002
XVAL: 0.54919804560331698 FVAL: -0.95613236012436187
XVAL: 0.54715576626861484 FVAL: -0.53938783326935613
XVAL: 0.55021887673599956 FVAL: -0.99668173493936607
SIMPLEX MINIMIZATION HAS CONVERGED.
XVAL: 0.55021887673509462 FVAL: -0.99668173496676893
FCN= -0.9966817 FROM SIMPLEX STATUS=PROGRESS 10 CALLS 93 TOTAL
EDM= 0.27E-10 STRATEGY= 0 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 para1 0.55022 0.25534E-01 0.18190E-11 0.32640E+07
IMPROVE HAS FOUND A TRULY NEW MINIMUM
*************************************
XVAL: 0.54819664405508539 FVAL: -0.79223514088950409
XVAL: 0.55224027991553515 FVAL: -0.69283908038410313
XVAL: 0.54227038041339182 FVAL: 0.34453274458608346
XVAL: 0.54971424097312460 FVAL: -0.99434925246263506
MIGRAD FAILS WITH STRATEGY=0. WILL TRY WITH STRATEGY=1.
XVAL: 0.55021887673509462 FVAL: -0.99668173496676893
XVAL: 0.54819664405508539 FVAL: -0.79223514088950409
XVAL: 0.55224027991553515 FVAL: -0.69283908038410313
XVAL: 0.54920786199518812 FVAL: -0.95718257528502082
XVAL: 0.55122968409989481 FVAL: -0.89911112206792232
XVAL: 0.55001669024027922 FVAL: -0.99998067325751328
XVAL: 0.55042105493490279 FVAL: -0.98776139504853400
XVAL: 0.54997584747860118 FVAL: -0.99995955985551355
XVAL: 0.54795351745711929 FVAL: -0.73849148324188363
XVAL: 0.55199735201488953 FVAL: -0.74968815036674952
XVAL: 0.54999624773645450 FVAL: -0.99999902563190934
XVAL: 0.54797392586790306 FVAL: -0.74316297267908471
XVAL: 0.55201774378283996 FVAL: -0.74505962542737447
MIGRAD MINIMIZATION HAS CONVERGED.
FCN= -0.9999990 FROM MIGRAD STATUS=CONVERGED 17 CALLS 110 TOTAL
EDM= 0.87E-06 STRATEGY=1 ERROR MATRIX UNCERTAINTY= 5.4%
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 para1 0.55000 0.42119E-02 -0.41006E-04 0.23334
找最小值有一定随机性，例如scale改为10，结果如下：
XVAL: 0.50000000000000000 FVAL: 0.97560974474682971
**********
** 1 **SET PRINTOUT 0.1587-313
**********
PARAMETER DEFINITIONS:
NO. NAME VALUE STEP SIZE LIMITS
1 'para1 ' 0.50000 0.10000E-02 0.0000 1.0000
**********
** 2 **SET STRATEGY 0.5305-314
**********
**********
** 3 **MIGRAD 0.7703E-26 0.000
**********
XVAL: 0.50000000000000000 FVAL: 0.97560974474682971
XVAL: 0.50001000000696405 FVAL: 0.97561443853647745
XVAL: 0.49998999999303595 FVAL: 0.97559542035102087
XVAL: 0.50000312758495857 FVAL: 0.97561224778513467
XVAL: 0.49999687241504148 FVAL: 0.97560629966369106
XVAL: 0.27112785285767249 FVAL: -0.52758163940408165
XVAL: 0.15398026789026997 FVAL: -0.12261175677271749
XVAL: 1.1989409613079260E-002 FVAL: -0.20823130360913300
XVAL: 0.97285414107188140 FVAL: -0.22389160020031401
XVAL: 0.64351971292093135 FVAL: 0.41239283833208851
XVAL: 0.45398478997104891 FVAL: -0.28708891011905729
XVAL: 1.4714439555999104E-002 FVAL: -2.3178262258938803E-002
XVAL: 0.10786339105596815 FVAL: -0.26500661254163782
XVAL: 0.27115566015938020 FVAL: -0.52590238908331055
XVAL: 0.27110004645147623 FVAL: -0.52922028070892424
XVAL: 0.27113157702186796 FVAL: -0.52735909099236555
XVAL: 0.27112412870953984 FVAL: -0.52780345940350271
MINUIT WARNING IN MIGRAD
============== Negative diagonal element 1 in Error Matrix
MINUIT WARNING IN MIGRAD
============== 0.10E+01 added to diagonal of error matrix
XVAL: 3.9654383933650061E-002 FVAL: 0.27579520981445249
XVAL: 0.13870528071376459 FVAL: 0.34121009693205140
XVAL: 0.79262987213766822 FVAL: -0.42655992445560387
XVAL: 7.3326929957164055E-002 FVAL: -0.15333568605379091
XVAL: 1.9534839593589814E-002 FVAL: 0.25940187431116379
XVAL: 0.11829220711988486 FVAL: 0.30013223382948379
XVAL: 0.19440481488797012 FVAL: -8.2091329835333349E-002
XVAL: 0.23490668896173283 FVAL: -1.4707140072779509E-002
XVAL: 0.25622121357561078 FVAL: 0.20091366107442823
XVAL: 0.26698731430451272 FVAL: -0.32458081316210441
XVAL: 0.26998820979165705 FVAL: -0.56051350889646478
XVAL: 0.26999196888929766 FVAL: -0.56052218289917344
XVAL: 0.26998445071050720 FVAL: -0.56050405333887499
XVAL: 0.27003239936882417 FVAL: -0.56056605747890120
XVAL: 0.27003606756336546 FVAL: -0.56056556321380246
XVAL: 0.27002873118998105 FVAL: -0.56056580721666105
XVAL: 0.27003179813580114 FVAL: -0.56056606747403148
XVAL: 0.27003546536332806 FVAL: -0.56056569543803192
XVAL: 0.27002813092396422 FVAL: -0.56056569537850387
MIGRAD MINIMIZATION HAS CONVERGED.
FCN= -0.5605661 FROM MIGRAD STATUS=CONVERGED 35 CALLS 36 TOTAL
EDM= 0.12E-11 STRATEGY= 0 NO ERROR MATRIX
EXT PARAMETER CURRENT GUESS STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 para1 0.27003 0.10000E-02 -0.13542E-05 -0.36034E-05
**********
** 4 **IMPROVE 0.7703E-26
**********
XVAL: 0.27003179813580114 FVAL: -0.56056606747403148
XVAL: 0.27003546536332806 FVAL: -0.56056569543803192
XVAL: 0.27002813092396422 FVAL: -0.56056569537850387
START ATTEMPT NO. 1 TO FIND NEW MINIMUM
XVAL: 0.26715423783747744 FVAL: -0.34793129531242889
XVAL: 0.26884959940495695 FVAL: -0.52243692970045741
XVAL: 0.26546228777173131 FVAL: -7.9974662997188098E-002
XVAL: 0.26377377399704915 FVAL: 0.20655223051064761
XVAL: 0.26040715422610572 FVAL: 0.53943685564288935
XVAL: 0.25705457582584773 FVAL: 0.32374341260153544
XVAL: 0.25039232821619484 FVAL: -0.52297048735803675
XVAL: 0.24378859291284788 FVAL: 0.19170232312021507
XVAL: 0.23076283725979169 FVAL: -0.48110980191819763
XVAL: 0.21798952199030558 FVAL: 0.38390087771213177
XVAL: 0.19324787040979696 FVAL: -0.23015053025528320
XVAL: 0.16965640884112038 FVAL: -0.40634957961240870
XVAL: 0.12627324312217220 FVAL: -0.13930527561428729
XVAL: 0.14730359523075209 FVAL: -0.25257407722609548
XVAL: 0.16965640884112038 FVAL: -0.40634957961240870
XVAL: 0.15477467242351778 FVAL: -2.7606512305086924E-002
XVAL: 0.18507665929375566 FVAL: -1.0328123415746682E-002
XVAL: 0.16214668910348046 FVAL: 0.31189518473251643
XVAL: 0.16965640884112038 FVAL: -0.40634957961240870
XVAL: 0.16008159187939547 FVAL: 0.39663777711190262
XVAL: 0.17945121037716505 FVAL: 0.41514883406712205
XVAL: 0.16484110029697024 FVAL: 2.0092063616358898E-002
XVAL: 0.17452671581800311 FVAL: -6.1473245568654404E-002
XVAL: 0.16724182944290883 FVAL: -0.26267130112242415
XVAL: 0.17208473799059004 FVAL: -0.32661401778953603
XVAL: 0.17086886100066601 FVAL: -0.39508286880155324
XVAL: 0.16844739412833143 FVAL: -0.35970929631772686
XVAL: 0.17026220602773540 FVAL: -0.40811047932895633
XVAL: 0.16905147101675339 FVAL: -0.38998127720378378
XVAL: 0.16995920011259030 FVAL: -0.40907977770651133
XVAL: 0.16995920011259030 FVAL: -0.40907977770651133
START SIMPLEX MINIMIZATION. CONVERGENCE WHEN EDM .LT. 0.10E+00
XVAL: 0.17601314954341457 FVAL: 0.13046616297364058
XVAL: 0.16756120971072003 FVAL: -0.29260737413489657
XVAL: 0.17092218996537145 FVAL: -0.39323972163338716
XVAL: 0.16899837492565162 FVAL: -0.38786515745248218
XVAL: 0.17044042485017086 FVAL: -0.40581031250609451
XVAL: 0.16947851654169227 FVAL: -0.40303262782579863
XVAL: 0.17019974488484091 FVAL: -0.40861366423810264
SIMPLEX MINIMIZATION HAS CONVERGED.
XVAL: 0.16995920011259036 FVAL: -0.40907977770651149
FCN= -0.4090798 FROM SIMPLEX STATUS=PROGRESS 8 CALLS 78 TOTAL
EDM= 0.17E-15 STRATEGY= 0 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 para1 0.16996 0.60119E-02 0.11102E-15 1325.2
FCN= -0.5605661 FROM SIMPLEX STATUS=RESET 8 CALLS 78 TOTAL
EDM= 0.12E-14 STRATEGY=0 ERROR MATRIX UNCERTAINTY= 50.0%
EXT PARAMETER APPROXIMATE INTERNAL INTERNAL
NO. NAME VALUE ERROR STEP SIZE VALUE
1 para1 0.27003 0.60119E-02 0.60119E-04 -0.47792
结论minuit不适合给单个可调参数找最小值，很容易找到极小值，最好将初始值设置在最小值同一个山谷内