# Mathematica:FindRoot::jsing: Encountered a singular Jacobian at the point {d} = … when solving an equation

I got a lot of error message when running the following codes:

prdiff[dp_, group_] := NIntegrate[2*2*1/(2*Pi) Cos[delta*t]*Product[Sin[c*Q[[group]]^i*t]/(c*Q[[group]]^i*t), {i, k + 1,k + h}]/c*Product[UnitStep[c - Abs[cn[[i, group]]]]/(2*c), {i, 3, k + 1}], {t, 0.0000001, Infinity}, {c, 0.0000001, Infinity}, {delta,0.0000001, dp}]/Integrate[1/c*Product[UnitStep[c - Abs[cn[[i, group]]]]/(2*c), {i, 3, k + 1}], {c, 0.0000001, Infinity}] - p
dListA = Table[FindRoot[prdiff[d, group], {d, d /. dListC[[group, 1]]}], {group, 1,16}]


where

Q={0.374823, 0.472246, 0.540587, 0.594993, 0.640938, 0.681097,0.717011, 0.749645, 0.779662, 0.807531, 0.833598, 0.858129,0.881331,0.903376, 0.924391, 0.944492}
k=2
h=10
dListC={{d -> 1.05872}, {d -> 2.32267}, {d -> 3.42827}, {d -> 4.32914}, {d ->5.14583}, {d -> 5.86437}, {d -> 6.50946}, {d -> 7.13946}, {d -> 7.77764}, {d -> 8.35564}, {d -> 9.03415}, {d -> 9.68592}, {d ->10.4552}, {d -> 11.2388}, {d -> 12.1519}, {d -> 13.1069}}
p=0.9544
cn={{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1.33739, -1.39482, -1.30994, -1.18525, -1.07646, -0.975324,-0.883788, -0.806052, -0.739899, -0.67623, -0.626402,-0.578394,-0.539806, -0.503205, -0.47299, -0.444406}, {-0.451706, -0.392128,-0.358641, -0.383565, -0.423396, -0.466082, -0.53215,-0.60404,-0.653776, -0.714855, -0.783097, -0.845636, -0.913242, -0.951801,-0.993216, -1.09363}, {0.646079, 0.567603, 0.491802, 0.382459, 0.288999, 0.150248, 0.0691451, -0.000652212, -0.115531, -0.177339, -0.259667,-0.308962, -0.473129, -0.633472, -0.87269, -1.56666}, {1.77469, 2.03892, 2.53728, 2.71311, 2.98287, 3.1305, 3.18962, 3.21899, 3.28681, 3.2669, 3.29087, 3.26718, 3.3841, 3.45476, 3.61692, 4.32612}}


the error messages are:

NIntegrate::nlim: delta = d is not a valid limit of integration.

NIntegrate::nlim: delta = d is not a valid limit of integration.

NIntegrate::nlim: delta = d is not a valid limit of integration.

General::stop: Further output of NIntegrate::nlim will be suppressed during this calculation.

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.36070001885559877 and 0.0020873226492962383 for the integral and error estimates.

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

General::stop: Further output of NIntegrate::izero will be suppressed during this calculation.

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.36070001885559877 and 0.0020873226492962383 for the integral and error estimates.

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

General::stop: Further output of NIntegrate::slwcon will be suppressed during this calculation.

NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.36070001964905846 and 0.002087321948358613 for the integral and error estimates.

General::stop: Further output of NIntegrate::eincr will be suppressed during this calculation.

Divide::infy: Infinite expression -(0.0103937/0.) encountered.

Divide::infy: Infinite expression -(0.0103937/0.) encountered.

Divide::infy: Infinite expression -(0.0103937/0.) encountered.

General::stop: Further output of Divide::infy will be suppressed during this calculation.

FindRoot::jsing: Encountered a singular Jacobian at the point {d} = {1.05872}. Try perturbing the initial point(s).

FindRoot::jsing: Encountered a singular Jacobian at the point {d} = {2.32267}. Try perturbing the initial point(s).

FindRoot::jsing: Encountered a singular Jacobian at the point {d} = {3.42827}. Try perturbing the initial point(s).

General::stop: Further output of FindRoot::jsing will be suppressed during this calculation.

NIntegrate::inumr: The integrand (2.78145*10^11 Cos[1.1397 d] Sin[0.360095 c] <<8>> Sin[0.24526 (0. +c)] UnitStep[-0.975324+c])/(0. +c)^12 has evaluated to non-numerical values for all sampling points in the region with boundaries {{\[Infinity],1.}}.

NIntegrate::inumr: The integrand (2.78145*10^11 Cos[1.1397 d] Sin[0.360095 c] <<8>> Sin[0.24526 (0. +c)] UnitStep[-0.975324+c])/(0. +c)^12 has evaluated to non-numerical values for all sampling points in the region with boundaries {{\[Infinity],1.}}.

NIntegrate::inumr: The integrand (2.78145*10^11 Cos[1.1397 d] Sin[0.360095 c] <<8>> Sin[0.24526 (0. +c)] UnitStep[-0.975324+c])/(0. +c)^12 has evaluated to non-numerical values for all sampling points in the region with boundaries {{\[Infinity],1.}}.

General::stop: Further output of NIntegrate::inumr will be suppressed during this calculation.


it seems that my Integrand function has some singularity, but I couldn't find them since all integral intervals do not contain 0.

Update:I find that the most important error is

Divide::infy: Infinite expression -(0.0103937/0.) encountered.


Because I find that if prdiff[d,group]=x, then the error shows

Divide::infy: Infinite expression -(x/0.) encountered.


For example, if I try

prdiff[1, 1]


I get

0.00822213


then I try

FindRoot[prdiff[d, 1] == 0, {d, 1}]


then the error message says

Divide::infy: Infinite expression -(0.00822213/0.) encountered.


It is very strange because there is no expression like prdiff(d,group)/0. And since the prdiff function could give a normal value, the error should not be in the NIntegrate or Integrate function and may in the FindRoot.How should I do?

• Please show all the code needed -- you are missing definitions for cn[[ ]]. Next, break apart the code and see which piece is giving the problem. For instance, you have an Integrate and an NIntegrate -- which is giving the error messages? Commented Mar 24, 2018 at 14:01
• I have edited the question. Thanks for your remind. Commented Mar 24, 2018 at 15:14
• When you do prdiff[1,1] you do not get 0.00822213. You get "NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small." followed by the number 0.00822213. So look at your NIntegrate. Commented Mar 24, 2018 at 16:23
• But what about the "Divide::infy:" error? It shows only when FindRoot is used. Although I get some messages when running prdiff[1,1], the result is correct. I couldn't find any problems in my NIntegrate even after many checks. Commented Mar 24, 2018 at 16:44
• I usually start work on the first error, not the 20th or so. Often one error leads to another. Therefore your choice for the title seems misleading. Commented Mar 24, 2018 at 20:20

The principal problem is the need for _?NumericQ on the variables of prdiff, which probably makes this a duplicate of What are the most common pitfalls awaiting new users?. (For one of the variables, I used the more specific _Integer?Positive instead. I also precomputed the Integrate[..] factors for a modest speed up. The whole takes a very, very long time, due to the multiple calls to a 3D NIntegrate.)

iTab = Table[
Integrate[
1/c*Product[
UnitStep[c - Abs[cn[[i, group]]]]/(2*c), {i, 3, k + 1}], {c,
0.0000001, Infinity}],
{group, 1, 16}];

prdiff[dp_?NumericQ, group_Integer?Positive] :=
NIntegrate[
2*2*1/(2*Pi) Cos[delta*t]*
Product[Sin[c*Q[[group]]^i*t]/(c*Q[[group]]^i*t), {i, k + 1,
k + h}]/c*
Product[UnitStep[c - Abs[cn[[i, group]]]]/(2*c), {i, 3,
k + 1}], {t, 0.0000001, Infinity}, {c, 0.0000001,
Infinity}, {delta, 0.0000001, dp}]/iTab[[group]] - p
Monitor[
dListA =
Table[FindRoot[prdiff[d, group], {d, d /. dListC[[group, 1]]},
PrecisionGoal -> 4,   (* b/c of NIntegrate, probably can't get high precision *)
AccuracyGoal -> 3],
{group, 1, 16}],
{d, group}]

(* some time later... *)
dListA
(*  {{d -> 0.81095}, ..., {d -> 9.55454}}  *)

• It works! Thank you very much! Commented Mar 25, 2018 at 8:55
• @Peiyu_WEI You're welcome. Commented Mar 25, 2018 at 9:54