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locating overlow/underflow

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112176] locating overlow/underflow
  • From: Leslaw Bieniasz <nbbienia at cyf-kr.edu.pl>
  • Date: Fri, 3 Sep 2010 06:10:12 -0400 (EDT)


Hi,

I am not very much familiar with MATHEMATICA, but
I have to tabulate a certain mathematical expression in a possibly large
range of independent variable, and with the highest possible number of
accurate digits. I have a notebook file that does the job (included
below in text form, together with the results) but during the cell 
evaluation I obtain error messages stating that underflow or overflow occurred.
My question is: is there any way to rewrite the code in such a way so 
that I can obtain an information where exactly (that is for which values 
of independent variable) the errors actually occur?
Or, is there any way to rewrite the code or change the program settings
in such a way so that the errors do not occur at all?
I am using MATHEMATICA 6.0.

One more question: I also need a special function representing
the so-called Dawson integral (related to error functions).
Is such a function available in MATHEMATICA?

Leslaw

-----------------

In[1]:= Table[{N[y],
   SetPrecision[
    2 ((y - 1) Exp[y] Erfc[Sqrt[y]] + (y/3 - 1) 2 Sqrt[y/Pi] +
        1)/(y y), 70],
   Precision[
    2 ((y - 1) Exp[y] Erfc[Sqrt[y]] + (y/3 - 1) 2 Sqrt[y/Pi] +
        1)/(y y)]
   },
  {y,  {
    1/10^19, 2/10^19, 5/10^19,
    1/10^18, 2/10^18, 5/10^18,
    1/10^17, 2/10^17, 5/10^17,
    1/10^16, 2/10^16, 5/10^16,
    1/10^15, 2/10^15, 5/10^15,
    1/10^14, 2/10^14, 5/10^14,
    1/10^13, 2/10^13, 5/10^13,
    1/10^12, 2/10^12, 5/10^12,
    1/10^11, 2/10^11, 5/10^11,
    1/10^10, 2/10^10, 5/10^10,
    1/10^9,  2/10^9,    5/10^9,
    1/10^8,  2/10^8,    5/10^8,
    1/10^7,  2/10^7,    5/10^7,
    1/10^6,  2/10^6,    5/10^6,
    1/10^5,  2/10^5,    5/10^5,
    1/10^4,  2/10^4,    5/10^4,
    1/10^3,  2/10^3,    5/10^3,
    1/10^2,  2/10^2,    5/10^2,
    1/10^1,  2/10^1,    5/10^1,
    10^0,        2 10^0,    5 10^0,
    10^1,        2 10^1,    5 10^1,
    10^2,        2 10^2,    5 10^2,
    10^3,        2 10^3,    5 10^3,
    10^4,        2 10^4,    5 10^4,
    10^5,        2 10^5,    5 10^5,
    10^6,        2 10^6,    5 10^6,
    10^7,        2 10^7,    5 10^7,
    10^8,        2 10^8,    5 10^8,
    10^9,        2 10^9,    5 10^9,
    10^10,      2 10^10, 5 10^10,
    10^11,      2 10^11, 5 10^11,
    10^12,      2 10^12, 5 10^12,
    10^13,      2 10^13, 5 10^13,
    10^14,      2 10^14, 5 10^14,
    10^15,      2 10^15, 5 10^15

    }}]
TableForm[%]


During evaluation of In[1]:= General::ovfl: Overflow occurred in computation. >>
During evaluation of In[1]:= General::unfl: Underflow occurred in computation. >>
During evaluation of In[1]:= General::ovfl: Overflow occurred in computation. >>
During evaluation of In[1]:= General::unfl: Underflow occurred in computation. >>
During evaluation of In[1]:= General::ovfl: Overflow occurred in computation. >>
During evaluation of In[1]:= General::stop: Further output of General::ovfl will be suppressed \
during this calculation. >>
During evaluation of In[1]:= General::unfl: Underflow occurred in computation. >>
During evaluation of In[1]:= General::stop: Further output of General::unfl will be suppressed \
during this calculation. >>
Out[1]= {{1.*10^-19,
   0.9999999997145401414822232883268, \[Infinity]}, {2.*10^-19,
   0.9999999995962987966101077255753, \[Infinity]}, {5.*10^-19,
   0.99999999936169235169104104847744, \[Infinity]}, {1.*10^-18,
   0.999999999097296666990256607119881, \[Infinity]}, {2.*10^-18,
   0.999999998723384704048748762709682, \[Infinity]}, {5.*10^-18,
   0.9999999979814939857172052899296470, \[Infinity]}, {1.*10^-17,
   0.99999999714540142082223286981099390, \[Infinity]}, {2.*10^-17,
   0.99999999596298797810107721769016053, \[Infinity]}, {5.*10^-17,
   0.999999993616923546910410334316185418, \[Infinity]}, {1.*10^-16,
   0.9999999909729667299025656456386676196, \[Infinity]}, {2.*10^-16,
   0.9999999872338471604874864234309767634, \[Infinity]}, {5.*10^-16,
   0.99999997981494015717204814138949104458, \[Infinity]}, {1.*10^-15,
   0.999999971454014808222315240716855446413, \[Infinity]}, {2.*10^-15,
    0.999999959629880981010734113646272096829, \[Infinity]}, \
{5.*10^-15,
   0.9999999361692384691039528849366322454328, \[Infinity]}, \
{1.*10^-14,
   0.99999990972967329902523089626863231557756, \[Infinity]}, \
{2.*10^-14,
   0.99999987233848360487366056855794216146181, \[Infinity]}, \
{5.*10^-14,
   0.999999798149431571715723507477761640655816, \[Infinity]}, \
{1.*10^-13,
   0.9999997145402080822096950163327874248582992, \[Infinity]}, \
{2.*10^-13,
   0.9999995962989298100692778901205287332143837, \[Infinity]}, \
{5.*10^-13,
   0.99999936169268469088907068033811563428191196, \[Infinity]}, \
{1.*10^-12,
   0.999999097297332989826749069417310407519066954, \[Infinity]}, \
{2.*10^-12,
   0.999998723386036047532940832853199726877995106, \[Infinity]}, \
{5.*10^-12,
   0.9999979814973157123993342769964108375782037344, \[Infinity]}, \
{1.*10^-11,
   0.99999714540808080863958180501897178805571422797, \[Infinity]}, \
{2.*10^-11,
   0.99999596300129806262962246877623843211942468409, \[Infinity]}, \
{5.*10^-11,
   0.999993616956846758433099710334117289408922294795, \[Infinity]}, \
{1.*10^-10,
   0.9999909730333294727098451619220974886096584308203, \[Infinity]}, \
{2.*10^-10,
   0.9999872339803592716735465274233676888555754834793, \[Infinity]}, \
{5.*10^-10,
   0.99997981527315236614873799872200298660421599112079, \[Infinity]}, \
{1.*10^-9,
   0.999971454680794629252230555027937586208375206273895, \
\[Infinity]}, {2.*10^-9,
   0.999959631212942564038868577810708150105882262021567, \
\[Infinity]}, {5.*10^-9,
   0.9999361725683171323430450418935382232030480963633574, \
\[Infinity]}, {1.*10^-8,
   0.99990973633286919192909560231769338552621527844005465, \
\[Infinity]}, {2.*10^-8,
   0.99987235180238915077617597090031017455487299571323293, \
\[Infinity]}, {5.*10^-8,
   0.999798182726766378746112583033893508505423363651441785, \
\[Infinity]}, {1.*10^-7,
   0.9997146067944913962662443861381550812129566065971701181, \
\[Infinity]}, {2.*10^-7,
   0.9995964320913723716649649533249490727923551184325208368, \
\[Infinity]}, {5.*10^-7,
   0.99936202553277550583443780378864879173919378767298350956, \
\[Infinity]}, {1.*10^-6,
   0.999097962903381392618257918169808186133578565612905041807, \
\[Infinity]}, {2.*10^-6,
   0.998724716821223900132151301649431782912244457614617752179, \
\[Infinity]}, {5.*10^-6,
   0.9979848225159930290612974417133409969991182485096079390572, \
\[Infinity]}, {0.00001,
   0.99715205451245340984272072402376554746735097950171503191800, \
\[Infinity]}, {0.00002,
   0.99597628295012871025175184133716000186655058157832590725788, \
\[Infinity]}, {0.00005,
   0.993650105491543032373266408651022396609428946524109448862154, \
\[Infinity]}, {0.0001,
   0.9910392059578652166795069631112683822179123631575592632039850, \
\[Infinity]}, {0.0002,
   0.9873659744612733772583124002782906358336022836617031938910022, \
\[Infinity]}, {0.0005,
   0.98014352895111543381382637050340393731816757683943978354872738, \
\[Infinity]}, {0.001,
   0.972107333318279894078855918414228411392356971871252393881957589, \
\[Infinity]}, {0.002,
   0.960925741846674240019638473110302681634263594240453539524316569, \
\[Infinity]}, {0.005,
   0.9393566121127200498908648890246331681521689289144646538210050146, \
\[Infinity]}, {0.01,
   0.9159902009040924490250299590163413654635346369396778873520195889, \
\[Infinity]}, {0.02,
   0.8845489144448704427280817404061389113644987295066944272551439941, \
\[Infinity]}, {0.05,
   0.82723454417674794029046907957368671185022940780693788453724800037,\
  \[Infinity]}, {0.1,
   0.769748582788727344538491837757160790299606272079824730272354026215\
, \[Infinity]}, {0.2,
   0.699232242525341506949772819117642562815375532716177178907138434103\
7, \[Infinity]}, {0.5,
   0.588143259726577320679301839443053773222030917216494297935021239844\
33, \[Infinity]}, {1.,
   0.495494443872649901471788129171273104415864988456003048415771408771\
620, \[Infinity]}, {2.,
   0.402140480955548821133851740739495786058744699228268255193373584174\
60, \[Infinity]}, {5.,
   0.288911482041544026351044834721976387733419669000343513698745282192\
, \[Infinity]}, {10.,
   0.2172222401396003819699281665905984026748849752211952762443786524, \
\[Infinity]}, {20.,
   0.159682833889492164165276148953035239355107803536204179301335, \
\[Infinity]}, {50.,
   0.103898856438174453901130042254777276403821295925701001674896046826\
4487, \[Infinity]}, {100.,
   0.074280111128504166978597120002158527004820661741654346725511978095\
5073, \[Infinity]}, {200.,
   0.052840382044123199631462541766374044290452985187147781191204691936\
3881, \[Infinity]}, {500.,
   0.033548539386005934478949701602134509526379532531957500468075086230\
3655, \[Infinity]}, {1000.,
   0.023754585587166811032652812370733379184448439544392165425468126451\
3817, \[Infinity]}, {2000.,
   0.016808758359715603635767774844392236608100743804294132512009896293\
3174, \[Infinity]}, {5000.,
   0.010635348315159696919776357112721673103593889162191241340002282113\
68411, \[Infinity]}, {10000.,
   0.007521419232226881694285320256609641558690025685331124462521775358\
92807, \[Infinity]}, {20000.,
   0.005318836433152609806824908867860117033202568088113139797523807695\
85166, \[Infinity]}, {50000.,
   0.003364076567698290692034556074814691590610990159037322908916949765\
51381, \[Infinity]}, {100000.,
   0.002378796671852805656000459926141396168102735040314336789405930067\
58431, \[Infinity]}, {200000.,
   0.001682075782256212856024086228569462951010275705207774046173766787\
05007, \[Infinity]}, {500000.,
   0.001063842897522669330939755783504367122329615797119676515766230784\
462375, \[Infinity]}, {1.*10^6,
   0.000752251651682815386411359767278213838037055070919378369497456335\
765897, \[Infinity]}, {2.*10^6,
   0.000531922642092663962568285063847383265111997405192167429802686793\
128027, \[Infinity]}, {5.*10^6,
   0.000336417568757356848041333360766808363214193818969988566716986874\
026595, \[Infinity]}, {1.*10^7,
   0.000237883179824548473177870798184711650639486725296508177959537387\
265020, \[Infinity]}, {2.*10^7,
   0.000168208822190680446941020059970130276635534860948672447590320829\
019214, \[Infinity]}, {5.*10^7,
   0.000106384604916310375159711825221478563296699198272857096813639705\
4260506, \[Infinity]}, {1.*10^8,
   0.000075225276678188320905210654259981758159131350353428366909503619\
2342803, \[Infinity]}, {2.*10^8,
   0.000053192303654632073665159706174172569621308032205289040073382595\
7980036, \[Infinity]}, {5.*10^8,
   0.000033641766859351499459713022897197101554901804981600259808868366\
7584749, \[Infinity]}, {1.*10^9,
   0.000023788321513023132483940405903441743127938158022647462615222027\
1704841, \[Infinity]}, {2.*10^9,
   Indeterminate, \[Infinity]}, {5.*10^9,
   Indeterminate, \[Infinity]}, {1.*10^10,
   Indeterminate, \[Infinity]}, {2.*10^10,
   Indeterminate, \[Infinity]}, {5.*10^10,
   Indeterminate, \[Infinity]}, {1.*10^11,
   Indeterminate, \[Infinity]}, {2.*10^11,
   Indeterminate, \[Infinity]}, {5.*10^11,
   Indeterminate, \[Infinity]}, {1.*10^12,
   Indeterminate, \[Infinity]}, {2.*10^12,
   Indeterminate, \[Infinity]}, {5.*10^12,
   Indeterminate, \[Infinity]}, {1.*10^13,
   Indeterminate, \[Infinity]}, {2.*10^13,
   Indeterminate, \[Infinity]}, {5.*10^13,
   Indeterminate, \[Infinity]}, {1.*10^14,
   Indeterminate, \[Infinity]}, {2.*10^14,
   Indeterminate, \[Infinity]}, {5.*10^14,
   Indeterminate, \[Infinity]}, {1.*10^15,
   Indeterminate, \[Infinity]}, {2.*10^15,
   Indeterminate, \[Infinity]}, {5.*10^15, Indeterminate, \[Infinity]}}
Out[2]//TableForm= \!\(\*
TagBox[GridBox[{
{"1.`*^-19",
       "0.999999999714540141482223288326838720459494243711`31.\
397940007947568", "\[Infinity]"},
{"2.`*^-19",
       "0.999999999596298796610107725575341676095945272569`31.\
999999998975444", "\[Infinity]"},
{"5.`*^-19",
       "0.999999999361692351691041048477441625919324180724`32.\
79588001572411", "\[Infinity]"},
{"1.`*^-18",
       "0.999999999097296666990256607119880814097330501032`33.\
397940007054885", "\[Infinity]"},
{"2.`*^-18",
       "0.999999998723384704048748762709681849891326049256`33.\
999999996760074", "\[Infinity]"},
{"5.`*^-18",
       "0.999999997981493985717205289929647013080957505254`34.\
795880012221296", "\[Infinity]"},
{"1.`*^-17",
       "0.99999999714540142082223286981099389630332598802`35.\
39794000142732", "\[Infinity]"},
{"2.`*^-17",
       "0.999999995962987978101077217690160528669609472929`35.\
99999998975443", "\[Infinity]"},
{"5.`*^-17",
       "0.999999993616923546910410334316185417885058048876`36.\
79588000114441", "\[Infinity]"},
{"1.`*^-16",
       "0.999999990972966729902565645638667619594263768862`37.\
397939992500426", "\[Infinity]"},
{"2.`*^-16",
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99999996760067", "\[Infinity]"},
{"5.`*^-16",
       "0.9999999798149401571720481413894910445784508378646347837159`\
38.795879966116246", "\[Infinity]"},
{"1.`*^-15",
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39.397939936224944", "\[Infinity]"},
{"2.`*^-15",
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39.99999989754434", "\[Infinity]"},
{"5.`*^-15",
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40.795879855347465", "\[Infinity]"},
{"1.`*^-14",
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41.39793984695594", "\[Infinity]"},
{"2.`*^-14",
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41.999999676006816", "\[Infinity]"},
{"5.`*^-14",
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42.79587950506595", "\[Infinity]"},
{"1.`*^-13",
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43.39793928420148", "\[Infinity]"},
{"2.`*^-13",
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43.99999897544414", "\[Infinity]"},
{"5.`*^-13",
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44.79587839737977", "\[Infinity]"},
{"1.`*^-12",
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397938391512476", "\[Infinity]"},
{"2.`*^-12",
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45.99999676007544", "\[Infinity]"},
{"5.`*^-12",
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46.79587489458097", "\[Infinity]"},
{"1.`*^-11",
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{"2.`*^-11",
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47.99998975451427", "\[Infinity]"},
{"5.`*^-11",
       "0.9999936169568467584330997103341172894089222947947673053359871\
654236`48.795863817883124", "\[Infinity]"},
{"1.`*^-10",
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10575`49.397923837222464", "\[Infinity]"},
{"2.`*^-10",
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523977`49.999967601482766", "\[Infinity]"},
{"5.`*^-10",
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122079`50.7958287915341", "\[Infinity]"},
{"1.`*^-9",
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945939`51.397867565621986", "\[Infinity]"},
{"2.`*^-9",
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638353`51.99989755242632", "\[Infinity]"},
{"5.`*^-9",
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980834`52.79571804094228", "\[Infinity]"},
{"1.`*^-8",
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514809`53.3977783087853", "\[Infinity]"},
{"2.`*^-8",
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814713`53.999676087644175", "\[Infinity]"},
{"5.`*^-8",
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738418`54.79536794124357", "\[Infinity]"},
{"1.`*^-7",
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086583`55.39686788719281", "\[Infinity]"},
{"2.`*^-7",
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{"5.`*^-7",
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940789`56.794262070859546", "\[Infinity]"},
{"1.`*^-6",
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077127`57.39632446890655", "\[Infinity]"},
{"2.`*^-6",
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2644919383827241`57.99676813224104", "\[Infinity]"},
{"5.`*^-6",
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4559609081594948`58.790777354382435", "\[Infinity]"},
{"0.00001`",
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{"0.00002`",
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{"0.00005`",
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